master thesis absolute irreversibility in information ... · then, we review and derive...

Master Thesis

Absolute Irreversibility

in Information Thermodynamics

Yuto Murashita

February 2, 2015

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Abstract

Nonequilibrium equalities have attracted considerable interest in the context of statistical

mechanics and information thermodynamics. What is remarkable about nonequilibrium

equalities is that they apply to rather general nonequilibrium situations beyond the linear

response regime. However, nonequilibrium equalities are known to be inapplicable to some

important situations. In this thesis, we introduce a concept of absolute irreversibility as a

new class of irreversibility that encompasses the entire range of those irreversible situations

to which the conventional nonequilibrium equalities are inapplicable. In mathematical

terms, absolute irreversibility corresponds to the singular part of probability measure

and can be separated from the ordinary irreversible part by Lebesgue’s decomposition

theorem in measure theory. This theorem guarantees the uniqueness of the decomposition

of probability measure into singular and nonsingular parts, which enables us to give a well-

defined mathematical and physical meaning to absolute irreversibility. Consequently, we

derive a new type of nonequilibrium equalities in the presence of absolute irreversibility.

Inequalities derived from our nonequilibrium equalities give stronger restrictions on the

entropy production during nonequilibrium processes than the conventional second-law like

inequalities. Moreover, we present a new resolution of Gibbs’ paradox from the viewpoint

of absolute irreversibility. This resolution applies to a classical mesoscopic regime, where

two prevailing resolutions of Gibbs’ paradox break down.

Contents

1 Introduction 4

2 Review of Nonequilibrium Equalities 8

2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Discovery of fluctuation theorem . . . . . . . . . . . . . . . . . . . 8

2.1.2 Jarzynski equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3 Crooks fluctuation theorem . . . . . . . . . . . . . . . . . . . . . . 14

2.1.4 Further equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Unified formulation based on reference probabilities . . . . . . . . . . . . . 19

2.2.1 Langevin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Hamiltonian system . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Review of Information Thermodynamics 34

3.1 Maxwell’s demon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Original Maxwell’s demon . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.2 Szilard engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.3 Brillouin’s argument . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.4 Landauer’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Second law of information thermodynamics . . . . . . . . . . . . . . . . . . 40

3.2.1 Classical information quantities . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Second law under feedback control . . . . . . . . . . . . . . . . . . 44

3.2.3 Second laws of memories . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.4 Reconciliation of the demon with the conventional second law . . . 56

3.3 Nonequilibrium equalities under measurements and feedback control . . . . 57

3.3.1 Information-thermodynamic nonequilibrium equalities . . . . . . . . 57

3.3.2 Derivation of information-thermodynamic nonequilibrium equalities 59

3.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1

4 Nonequilibrium Equalities in Absolutely Irreversible Processes 64

4.1 Inapplicability of conventional integral nonequilibrium equalities and ab-

solute irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.1 Inapplicability of the Jarzynski equality . . . . . . . . . . . . . . . . 65

4.1.2 Definition of absolute irreversibility . . . . . . . . . . . . . . . . . . 67

4.2 Nonequilibrium equalities in absolutely irreversible processes . . . . . . . . 70

4.2.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.2 Physical implications . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Examples of absolutely irreversible processes . . . . . . . . . . . . . . . . . 75

4.3.1 Free expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.2 Process starting from a local equilibrium . . . . . . . . . . . . . . . 76

4.3.3 System with a trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Comparison with a conventional method . . . . . . . . . . . . . . . . . . . 81

5 Information-Thermodynamic Nonequilibrium Equalities in Absolutely

Irreversible Processes 84

5.1 Inforamtion-thermodynamic equalities . . . . . . . . . . . . . . . . . . . . 84

5.2 Unavailable information and associated equalities . . . . . . . . . . . . . . 88

5.3 Examples of absolutely irreversible processes . . . . . . . . . . . . . . . . . 90

5.3.1 Measurement and trivial feedback control . . . . . . . . . . . . . . . 90

5.3.2 Two-particle Szilard engine . . . . . . . . . . . . . . . . . . . . . . 93

5.3.3 Multi-particle Szilard engine . . . . . . . . . . . . . . . . . . . . . . 97

6 Gibbs’ Paradox Viewed from Absolute Irreversibility 101

6.1 History and Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1.1 Original Gibbs’ paradox . . . . . . . . . . . . . . . . . . . . . . . . 102

6.1.2 Gibbs’ paradox is not a paradox . . . . . . . . . . . . . . . . . . . . 104

6.1.3 Quantum resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.1.4 Pauli’s resolution based on the extensivity . . . . . . . . . . . . . . 110

6.2 Resolution from absolute irreversibility . . . . . . . . . . . . . . . . . . . . 113

6.2.1 Requirement and Results . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.2 Difference of the two processes . . . . . . . . . . . . . . . . . . . . . 114

6.2.3 Derivation of the factor N ! . . . . . . . . . . . . . . . . . . . . . . . 117

7 Conclusions and Future Prospects 120

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2

A From the Langevin Dynamics to Other Formulations 122

A.1 Path-integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.2 Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

B Measure Theory and Lebesgue’s Decomposition 127

B.1 Preliminary subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B.2 Classification of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

B.3 Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

B.4 Lebesgue’s decomposition theorem . . . . . . . . . . . . . . . . . . . . . . 130

3

Chapter 1

Introduction

In the mid-1990s, a significant breakthrough was achieved in the field of nonequilibrium

statistical physics. Evans, Cohen and Morris numerically found a new symmetry of the

probability distribution function of the entropy production rate in a steady-state shear-

driven flow [1]. This symmetry, later formulated in the form of fluctuation theorems,

was proven in chaotic systems by Gallavotti and Cohen [2], and later in various types of

systems [3–6]. In this way, the fluctuation theorems present a ubiquitous and universal

structure residing in nonequilibrium systems. What is important about the fluctuation

theorems is that they apply to systems far from equilibrium. Moreover, they can be

regarded as a generalized formulation of the linear response theory to rather general

nonequilibrium situations [7]. In this respect, the fluctuation theorems have attracted

considerable attention.

In the course of the development of the fluctuation theorems, an important relation was

found by Jarzynski [8, 9]. The Jarzynski equality or the integral nonequilibrium equality

relates the equilibrium free-energy difference between two configurations to an ensemble

property of the work performed on the system during a rather general nonequilibrium pro-

cess that starts from one of the two configurations. Thus, the Jarzynski equality enables

us to estimate the free-energy difference by finite-time measurements in a nonequilibrium

process outside of the linear response regime. Moreover, the Jarzynski equality concisely

reproduces the second law of thermodynamics and the fluctuation-dissipation relation in

the weak fluctuation limit. In this way, the integral nonequilibrium equality is a funda-

mental relation with experimental applications.

Another significant relation is the Crooks fluctuation theorem or the detailed nonequi-

librium equality [10, 11]. The Crooks fluctuation theorem compares the realization proba-

bility of a trajectory in phase space under a given dynamics with that of the time-reversed

trajectory under the time-reversed dynamics. The ratio of these two probabilities is ex-

actly quantified by the exponentiated entropy production. Therefore, irreversibility of

4

a path under time reversal is quantitatively characterized by the entropy production of

the path itself. Additionally, the Crooks fluctuation theorem succinctly reproduces the

Jarzynski equality.

Recently, the subject of nonequilibrium equalities marks a new development in the

field of feedback control. The theory of feedback control dates back to Maxwell [12]. In

his textbook of thermodynamics, Maxwell pointed out that the second law can be violated

if we have access to microscopic degrees of freedom of the system and illustrated the idea

in his renowned gedankenexperiment later christened by Lord Kelvin as Maxwell’s de-

mon. Maxwell’s demon is able to reduce the entropy of an isolated many-particle gas, by

measuring the velocity of the particles and manipulating them based on the information

of the measurement outcomes without expenditure of work. Maxwell’s demon triggered

subsequent discussions on the relations between thermodynamics and information pro-

cessing [13–15]. Although Maxwell’s demon has been a purely imaginary object for about

one and a half century, thanks to technological advances, the realization of Maxwell’s

demon in real experiments is now within our hands [16, 17]. Hence, the theory of feed-

back control has attracted considerable interest in these days, and quantitative relations

between thermodynamic quantities and information were established [18, 19], forming a

new field of information thermodynamics. One of the quantitative relation is known as

the second law of information thermodynamics, which states that the entropy reduction

in the feedback process is restricted by the amount of mutual information obtained in

the measurement process. As the Jarzynski equality is a generalization of the second law

of thermodynamics, the second law of information thermodynamics is generalized in the

form of the integral nonequilibrium equality [20]. Thus, nonequilibrium equalities are an

actively developing field in the context of information thermodynamics.

Despite of the wide applicability and interest of nonequilibrium equalities, they are

known to be inapplicable to free expansion [21, 22]. Moreover, information-thermodynamic

nonequilibrium equalities cannot apply to situations that involve such high-accuracy mea-

surements as error-free measurements [23]. This inapplicability roots from divergence of

the exponentiated entropy production, and can be circumvented at the level of the detailed

nonequilibrium equalities [24]. However, to obtain the corresponding integral nonequilib-

rium equalities, situation-specific modifications are needed, and moreover the form of the

obtained equalities is rather unusual. There has been no unified strategy to derive these

exceptional integral nonequilibrium equalities in the situations to which the conventional

integral nonequilibrium equalities cannot apply.

In this thesis, we propose a new concept of absolute irreversibility that constitutes

those irreversible situations to which the conventional integral nonequilibrium equalities

cannot apply. In physical terms, absolute irreversibility refers to those irreversible situa-

5

tions in which paths in the time-reversed dynamics do not have the corresponding paths

in the original forward dynamics, which makes stark contrast to ordinary irreversible situ-

ations, in which every time-reversed path has the corresponding original path. Therefore,

in the context of the detailed nonequilibrium equalities, irreversibility is so strong that the

entropy production diverges, whereas, in ordinary irreversible situations, irreversibility is

quantitatively characterized by a finite entropy production. In mathematical terms, ab-

solute irreversibility is characterized as the singular part of the probability measure in the

time-reversed dynamics with respect to the probability measure in the original dynamics.

Therefore, based on Lebesgue’s decomposition theorem in measure theory [25, 26], the

absolutely irreversible part is uniquely separated from the ordinary irreversible part. As

a result, we obtain nonequilibrium integral equalities that are applicable to absolutely

irreversible situations [27]. Furthermore, our nonequilibrium equalities give tighter re-

strictions on the entropy production than the conventional second-law like inequalities

[27, 28].

As an illustrative application of our integral nonequilibrium equalities and the notion

of absolute irreversibility, we consider the problem of gas mixing, which is what Gibbs’

paradox deals with [29]. Gibbs’ paradox is qualitatively resolved once we recognize the

equivocal nature of the thermodynamic entropy [30–32]. Although the standard quan-

titative resolution of Gibbs’ paradox in many textbooks is based on quantum statistical

mechanics, this resolution is indeed irrelevant to Gibbs’ paradox [31, 32]. Pauli gave

a correct quantitative analysis of Gibbs’ paradox based on the extensivity of the ther-

modynamic entropy [33, 32]. However, this resolution holds only in the thermodynamic

limit, and ignores any sub-leading effects. Based on our nonequilibrium equalities in the

presence of absolute irreversibility, we give a quantitative resolution of Gibbs’ paradox

that is applicable even to a classical mesoscopic regime, where sub-leading effects play an

important role.

This thesis is organized as follows. In Chap. 2, we briefly review history of nonequi-

librium equalities and a unified approach to derive them. In Chap. 3, we review a part

of historical discussions on Maxwell’s demon and some fundamental relations under mea-

surements and feedback control including the second law of information thermodynamics.

Then, we review and derive information-thermodynamic nonequilibrium equalities. In

Chaps. 4-6, we describe the main results of this study. In Chap. 4, we introduce a con-

cept of absolute irreversibility in an example of free expansion, to which the conventional

integral nonequilibrium equalities do not apply, and define absolute irreversibility in math-

ematical terms. Then, in situations without measurements and feedback control, we derive

nonequilibrium equalities in the presence of absolute irreversibility based on Lebesgue’s

decomposition theorem, and verify them in several illustrative examples. In Chap. 5, we

6

generalize the results in Chap. 4 to obtain information-thermodynamic nonequilibrium

equalities in the presence of absolute irreversibility and verify them analytically in a few

simple examples. In Chap. 6, we briefly review discussions and conventional resolutions

on Gibbs’ paradox and quantitatively resolve Gibbs’ paradox based on our nonequilibrium

equalities with absolute irreversibility. In Chap. 7, we summarize this thesis and discuss

some future prospects.

7

Chapter 2

Review of Nonequilibrium Equalities

In this chapter, we review nonequilibrium equalities in classical statistical mechanics.

Nonequilibrium equalities are exact equalities applicable to quite general nonequilibrium

systems, and are generalizations of the second law of thermodynamics and well-known

relations in linear response theory. Moreover, nonequilibrium equalities give one solution

of Loschmidt’s paradox.

In the former half of this chapter, we review various nonequilibrium equalities in

the chronological order. In the latter half, we review a unified method to derive the

nonequilibrium equalities introduced in the former half.

2.1 History

First of all, we briefly review history of nonequilibrium equalities.

2.1.1 Discovery of fluctuation theorem

The field of nonequilibrium equalities was initiated by Evans, Cohen, and Morris in 1993

[1]. In a steady state of thermostatted shear-driven flow, they numerically discovered a

novel symmetry in the probability distribution of the entropy production rate, which is

nowadays called the steady-state fluctuation theorem. The theorem reads

limt→∞

1

tln

P (Σ)

P (−Σ)= Σ, (2.1)

where t is the time interval and P (Σ) is the probability distribution function for the

time-averaged entropy production rate Σ in units of kBT , where kB is the Boltzmann

constant and T is the absolute temperature of the system (see Fig. 2.1). They justified

this symmetry by assuming a certain statistical ensemble of the nonequilibrium steady

8

(a) (b)

Figure 2.1: (a) Probability distribution of the entropy production rate obtained by nu-merical simulations in shear-driven flow. The abscissa shows the negative of the entropyproduction rate Σ in arbitrary units. We can observe that the probability of events vio-lating the second law does not vanish. (b) Demonstration of the steady-state fluctuationtheorem (2.1). The abscissa is the same as (a). The markers represent the logarithmof the ratio (log-ratio) of the probability of a positive entropy production to that of thereversed sign (namely the log-ratio in the left-hand side of Eq. (2.1)), and the dottedline represents the values of the log-ratio predicted by Eq. (2.1). We can confirm thatthe log-ratio is proportional to the negative of entropy production rate. Reproduced fromFigs. 1 and 2 in Ref. [1]. Copyright 1993 by the American Physical Society.

state. Subsequently, in the same system, a similar symmetry is disclosed in a transient

situation from the equilibrium state to the steady state [34]. It is known as the transient

fluctuation theorem and written as

P (Σ)

P (−Σ)= eΣt. (2.2)

This relation was proved under the same assumption as the steady-state fluctuation the-

orem (2.1), which suggests that the fluctuation theorem is not a property particular to

steady states, but applicable to wider classes of nonequilibrium situations. Equations

(2.1) and (2.2) demonstrate that the probability of a positive entropy production is ex-

ponentially greater than that of the reversed sign. Moreover, these fluctuation theorems

reproduce the Green-Kubo relation [35, 36] and Onsager’s reciprocity relation [37, 38] in

the limit of weak external fields [7]. Therefore, the fluctuation theorems can be regarded

as extensions of the well-established relations in the linear-response regime to a more

general nonequilibrium regime.

These fluctuation theorems (2.1) and (2.2) resolve Loschmidt’s paradox in the follow-

ing sense. Loschmidt’s paradox originates from his criticism to the H-theorem proposed

by Boltzmann, which demonstrates that the Shannon entropy of the probability distri-

bution function of phase space increases with time in a system obeying the Boltzmann

9

equation, although this equation is symmetric under time reversal. It was claimed that

the H-theorem is a derivation of the second law of thermodynamics, and the irreversible

macroscopic law can be derived from the reversible microscopic dynamics. However,

Loschmidt criticized this argument by the observation that it should be impossible to

deduce irreversible properties from the time-reversal symmetric dynamics. If we have a

process with a positive entropy production and reverse the velocity of all the particle in

the system at once, we can generate a process with a negative entropy production since

the dynamics is time-reversal symmetric. Therefore, the entropy of the system should not

always decrease, which is a defect of the H-theorem. Fluctuation theorems (2.1) and (2.2)

demonstrate that, as Loschmidt pointed out, paths with a negative entropy production

have nonzero probability. However, an important implication of the fluctuation theorems

is that the probability of a negative entropy production is exponentially suppressed in

large systems or in the long-time limit (namely when Σt 1). Therefore, paths violating

the second law cannot be observed in macroscopic systems in practice. In this way, the

fluctuation theorems reconcile Loschmidt’s paradox with the second law originating from

the reversible dynamics.

Soon after the discovery of the fluctuation theorems, the steady-state fluctuation the-

orem (2.1) was proved under the chaotic hypothesis, which is an extension of the ergodic

hypothesis, in dissipative reversible systems [2, 39]. Later, a proof free from the chaotic

hypothesis was proposed in Langevin systems since the Langevin dynamics is ergodic in

the sense that the system relaxes to the thermal equilibrium distribution in the long-time

limit [3], and then generalized to general Markov processes [4]. Moreover, both the steady-

state fluctuation theorem (2.1) and the transient fluctuation theorem (2.2) were shown in

general thermostatted systems in a unified manner [5]. It was pointed out that this proof

of the fluctuation theorems is applicable even to Hamiltonian systems, although it had

been believed that the thermostatting mechanism is needed for the fluctuation theorems

[6]. Thus, the fluctuation theorems are known to apply to wide classes of nonequilibrium

systems.

The transient fluctuation theorem (2.2) was experimentally demonstrated in Ref. [40].

They prepared a colloidal particle in an optical trap at rest, and then translated the trap

relative to the surrounding water. In this transient situation, they obtained the probability

distribution of entropy production, and observed trajectories of the particle violating the

second law at the level of individual paths (see Fig. 2.2). The amount of this violation

was confirmed to be consistent with Eq. (2.2). Later, the steady-state fluctuation theorem

(2.1) was also verified in the same setup [41].

10

(a) (b)

Figure 2.2: (a) Histograms of time-averaged entropy production of a dragged colloidalparticle in units of kBT , where kB is the Boltzmann constant. Two different types of thebars represent two different measurement-time intervals. We can observe that the proba-bility of paths violating the second law does not vanish. (b) Log-ratio of the probabilityof entropy production to that of its negative. The experimental results are consistentwith the transient fluctuation theorem (2.2). Reproduced from Figs. 1 and 4 in Ref. [40].Copyright 2002 by the American Physical Society.

2.1.2 Jarzynski equality

In 1997, Jarzynski discovered a remarkable exact nonequilibrium equality in a Hamiltonian

system [8]. Let H(λ, x) denote the Hamiltonian of the system, where λ is an external

parameter that we control to manipulate the system and x represents internal degrees of

freedom of the system. The system is initially in equilibrium with the inverse temperature

β, and we subject the system to a nonequilibrium process by our manipulation of λ from

λi to λf . Let F (λ) denote the equilibrium free energy of the system under a given external

parameter λ, i.e.,

e−βF (λ) =

∫dx e−βH(λ,x). (2.3)

The Jarzynski equality relates the free-energy difference ∆F := F (λf) − F (λi) to the

probability distribution of work W performed during the nonequilibrium process as

〈e−β(W−∆F )〉 = 1, (2.4)

where the angular brackets mean the statistical average under the initial equilibrium

state and the given nonequilibrium protocol. Soon after the discovery, the same equality is

proved in stochastic systems based on the master equation formalism [9]. It is noteworthy

that we assume nothing about how fast we change the external parameter λ, and therefore

the Jarzynski equality (2.4) remains valid under a rapid change of the parameter, which

11

means that the Jarzynski equality applies to processes beyond the linear response regime.

Moreover, the Jarzynski equality (2.4) is an extension of conventional thermodynamic

relations to the case of a rather general nonequilibrium regime [8]. First, it leads to a

second-law-like inequality in isothermal processes. Using Jensen’s inequality, we obtain

〈e−β(W−∆F )〉 ≥ e−β〈W−∆F 〉. (2.5)

Combining this inequality with the Jarzynski equality (2.4), we conclude

〈W 〉 ≥ ∆F, (2.6)

which is the second law of thermodynamics in isothermal processes. The equality con-

dition is that W has a single definite value, i.e., W does not fluctuate. We can regard

W − ∆F as the total entropy production of the system. Let ∆U and Q denote the

internal-energy difference and dissipated heat from the system to the heat bath, respec-

tively. Then, the first law of thermodynamics is ∆U = W − Q. We rewrite Eq. (2.6)

as

∆U −∆F

T+Q

T≥ 0, (2.7)

where T is the temperature of the heat bath (and therefore the initial temperature of

the system). The first term represents the entropy production of the system ∆S because

∆F = ∆U − T∆S, and the second term is the entropy production of the heat bath.

Therefore, Eq. (2.6) means that the entropy production of the total system must be

positive.

Secondly, the Jarzynski equality (2.4) reproduces the fluctuation-dissipation relation

in the linear response theory [8]. Let us denote σ = β(W − ∆F ), and assume that the

dissipated work W −∆F is much smaller than the thermal energy kBT , namely, |σ| 1.

Expanding ln〈e−σ〉 up to the second order in σ, we obtain

ln〈e−σ〉 ' −〈σ〉+1

2(〈σ2〉 − 〈σ〉2). (2.8)

Substituting the Jarzynski equality (2.4), we obtain

〈W 〉 −∆F =1

2kBT(〈W 2〉 − 〈W 〉2). (2.9)

The left-hand side is dissipation of the total system, and the right-hand side represents

fluctuations of the work during the process. This is one form of the fluctuation-dissipation

relation.

12

Not only does the Jarzynski equality reproduce the second law of thermodynamics,

but also it gives a stringent restriction on the probability of events violating the second

law [42]. Let σ0 be a positive number and let us calculate the probability that the entropy

production is smaller than −σ0:

Prob[σ ≤ −σ0] =

∫ −σ0−∞

dσ P (σ)

≤∫ −σ0−∞

dσ P (σ)e−σ−σ0

= 〈e−σ〉e−σ0

= e−σ0 , (2.10)

where we use the Jarzynski equality (2.4) to obtain the last line. Therefore, the probability

of negative entropy production is exponentially suppressed. While a negative entropy

production (σ < 0) may occasionally occur, a greatly negative entropy production (|σ| 1) is effectively prohibited by the Jarzynski equality.

In addition to the above-described properties, the Jarzynski equality (2.4) enables us

to determine the free-energy difference from our observation of how the system evolves

under a nonequilibrium process because

∆F = − 1

βln〈e−βW 〉, (2.11)

although the free-energy difference is an equilibrium property of the system [9, 43]. A

naive method to determine the free-energy difference in experiments or numerical simula-

tions is to conduct reversible measurements of work and use the fact W = ∆F . However,

this method is not realistic in general because an extremely long time is needed to real-

ize even approximately reversible processes. A more sophisticated method is to use the

linear response relation (2.9) to determine the free energy difference through the work

distribution obtained in the measurements. This method may still be time-consuming

because the manipulation must be slow enough for the system to remain in the linear

response regime. Equation (2.11) enables us to reduce the time of experiments or simu-

lations when we calculate the free-energy difference, because Eq. (2.11) is valid even for

rapid nonequilibrium processes. 1

Hummer and Szabo found an experimentally useful variant of the Jarzynski equality

[44]. The Hummer-Szabo equality can be utilized to rigorously reconstruct the free-energy

1The exact equality (2.11) requires more samples for convergence than the approximate equality (2.9)does [43]. Therefore, the total time required for convergence, which is the time of the process multipliedby the number of samples, can be longer when we use Eq. (2.11) than we use (2.9).

13

Figure 2.3: (a) Experimental setup of the stretching experiment. An RNA molecule isattached to two beads. One bead is in an optical trap to measure force and the other islinked to a piezoelectric actuator. (b) Difference between the estimated free energy andits actual value. The solid curves represent the reversible estimation (∆F = 〈W 〉). Thedotted curves represent the linear response estimation by Eq. (2.9). The dashed curvesrepresent the estimation based on the Jarzynski equality, namely, Eq. (2.11). We see thatthe Jarzynski equality gives the best estimation. Reproduced from Figs. 1 and 3 in Ref.[45]. Copyright 2002 by the American Association for the Advancement of Science.

landscape of a molecule from repeated measurements based, for example, on an atomic

force microscope or an optical tweezer. Shortly thereafter, in a setup with an optical

tweezer shown in Fig. 2.3 (a), the free-energy profile of a single molecule of RNA was

reconstructed by mechanically stretching the RNA in an irreversible manner [45] (see Fig.

2.3 (b)). This experiment demonstrated that the Jarzynski equality is useful in practice to

determine free-energy differences of systems. In a similar manner, the free-energy profile

of a protein was estimated by stretching the protein by an atomic force microscope [46].

2.1.3 Crooks fluctuation theorem

In 1998, Crooks offered a new proof of the Jarzynski equality in stochastic systems [47].

What is remarkable about this proof is the method comparing an original process with

the time-reversed process to derive nonequilibrium relations. One year later, this idea led

him to propose a novel relation now known as the Crooks fluctuation theorem [10], which

reads

P (σ)

P †(−σ)= eσ, (2.12)

14

where P (σ) is the probability distribution function of entropy production σ, and P †(σ)

is that in the time-reversed process. Later, this theorem was generalized to Hamiltonian

systems with multiple heat baths by Jarzynski [48].

The Crooks fluctuation theorem (2.12) can be regarded as a generalized version of

the steady-state fluctuation theorem (2.1) in systems symmetric with respect to reversal

of the perturbation that drives the steady flow; in this case, we have P †(−σ) = P (−σ),

which reduces Eq. (2.12) to Eq. (2.1), because there is no difference between the original

process and the time-reversed one. What is more, the Crooks fluctuation theorem easily

reproduces the Jarzynski equality as follows:

〈e−σ〉 =

∫ ∞−∞

dσ e−σP (σ)

=

∫ ∞−∞

dσ P †(σ)

= 1, (2.13)

where we have used the Crooks fluctuation theorem (2.12) to obtain the second line, and

the normalization of probability to obtain the last line. Moreover, the Crooks fluctuation

theorem implies that

σ = 0 ⇔ P (σ) = P †(−σ). (2.14)

Soon after, Crooks found a significant generalization of his own theorem [11]. He

related irreversibility of an individual path Γ in phase space to its own entropy production

σ[Γ], 2 that is,

P [Γ]

P†[Γ†] = eσ[Γ], (2.15)

where Γ† represents the time-reversed path of Γ; P is the path-probability functional

under a given dynamics, and P† is that under the time-reversed dynamics. To reproduce

Eq. (2.12) from Eq. (2.15), we note

P (σ) =

∫DΓP [Γ]δ(σ[Γ]− σ)

=

∫DΓP†[Γ†]eσ[Γ]δ(σ[Γ]− σ)

= eσ∫DΓ†P†[Γ†]δ(σ[Γ†] + σ)

= eσP †(−σ), (2.16)

2 We use box brackets to indicate that f [Γ] is a functional of Γ instead of a function.

15

where DΓ denotes the natural measure on the set of all paths, and we use Eq. (2.15)

to obtain the second line, and we assume entropy production is odd under time reversal,

namely σ[Γ] = −σ[Γ†], to obtain the third line. A more general form of the nonequilibrium

integral equality can be derived based on Eq. (2.15). Let F [Γ] be an arbitrary functional.

Then, we obtain

〈Fe−σ〉 =

∫DΓP [Γ]e−σ[Γ]F [Γ]

=

∫DΓ†P†[Γ†]F [Γ]

= 〈F〉†, (2.17)

where 〈· · · 〉† denotes the average over the time-reversed probability P†[Γ†]. When we set

F to unity, Eq. (2.17) reduces to Eq. (2.13). The idea to consider the path-probability

of an individual path is a crucial element to treat nonequilibrium equalities in a unified

manner as described in Sec. 2.2.

The Crooks fluctuation theorem (2.12) was experimentally verified [49] in a similar

setup in Ref. [45], which was used to verify the Jarzynski equality. The work extracted

when an RNA is unfolded and refolded was measured, and the work distribution of the

unfolding process and that of the refolding process, which is the time reversed process of

unfolding, were obtained (see Fig. 2.4 (a)). It was verified that the two distributions are

consistent with the Crooks fluctuation theorem (2.12) (see Fig. 2.4 (b)). Moreover, the

free-energy difference of a folded RNA and an unfolded one was obtained using Eq. (2.14),

and the obtained value is consistent with a numerically obtained one.

2.1.4 Further equalities

In this section, we will briefly review further nonequilibrium equalities. To this end, let

us introduce some kinds of entropy production. Mathematical definitions of these kinds

of entropy production are presented in Sec. 2.2.

The total entropy production ∆stot is the sum of the Shannon entropy production of

the system ∆s and the entropy production of the heat bath ∆sbath, that is,

∆stot = ∆s+ ∆sbath. (2.18)

The entropy production of the bath is related to the heat q dissipated from the system to

the bath

∆sbath = q/T, (2.19)

16

(b)(a)

Figure 2.4: (a) Work distribution of the unfolding process (solid lines) and the refoldingprocess (dashed lines). Three different colors correspond to three different speed of un-folding and refolding. We find that the work value where two distributions at the samespeed cross each other is independent of the speed (as shown by the vertical dotted line).Equation (2.14) implies that this value corresponds to zero entropy production, namely,the work value is equal to the free-energy difference. (b) Verification of the Crooks fluctu-ation theorem (2.12). In the setup of this experiment, the entropy production σ reducesto work W minus the free-energy difference. It is illustrated that the log-ratio of theprobability distributions of work is proportional to work itself. Reproduced from Fig. 2and the inset of Fig. 3 in Ref. [49]. Copyright 2005 by the Nature Publishing Group.

where T is the absolute temperature of the bath. In a steady-state situation, the heat q

is split into two parts as

q = qhk + qex, (2.20)

where qhk is called the housekeeping heat, which is the inevitable heat dissipation to

maintain the corresponding nonequilibrium steady state, and qex is called the excess heat,

which arises due to a non-adiabatic change of the external control parameter. Along with

these definitions of heat, we define two kinds of entropy production as

∆shk = qhk/T, ∆sex = qex/T. (2.21)

Hatano-Sasa relation

The Hatano-Sasa relation is a generalization of the Jarzynski equality [50]. The Jarzynski

equality relates the free-energy difference of two equilibrium states to the nonequilibrium

average starting from one of the equilibrium states. In a similar manner, the Hatano-Sasa

relation relates the difference of nonequilibrium potentials φ of two nonequilibrium steady

states, which is a generalization of U −F in equilibrium situations, to an average starting

17

from one of the steady state. The Hatano-Sasa relation reads

〈e−∆φ−∆sex/kB〉 = 1. (2.22)

From Jensen’s inequality, we obtain

〈∆φ〉+〈∆sex〉kB

≥ 0. (2.23)

This inequality can be interpreted as a nonequilibrium version of the Clausius inequality

(∆s ≥ −q/T ). In fact, the inequality (2.23) can be rewritten as

kB〈∆φ〉 ≥ −〈qex〉T

. (2.24)

The equality can be achieved in quasi-static transitions between the two nonequilibrium

steady states, which is also analogous to the Clausius inequality, whose equality is also

achieved in quasi-static transitions.

Seifert relation

The Seifert relation applies to an arbitrary nonequilibrium process starting from an arbi-

trary initial state [51]. The integral Seifert relation is given by

〈e−∆stot/kB〉 = 1. (2.25)

The corresponding inequality is

〈∆stot〉 ≥ 0, (2.26)

which can be considered as a second law in a nonequilibrium process. The detailed version,

which is analogous to the Crooks fluctuation theorem, is given by

P (−∆stot)

P (∆stot)= e−∆stot/kB . (2.27)

It is noteworthy that these relations hold for an arbitrary time interval. Equation (2.27)

can be regarded as a refinement of the steady state fluctuation theorem (2.1). The steady

state fluctuation theorem (2.1) holds only in the long-time limit. This is because Σ in

Eq. (2.1) is in fact the entropy production rate of the bath and does not include that of

the system. Therefore, in Eq. (2.1), the time interval should be long enough to ignore the

entropy production of the system compared with that of the bath.

18

Relation for housekeeping entropy production

In Ref. [52], a relation for housekeeping entropy production is also obtained as

〈e−∆shk/kB〉 = 1, (2.28)

which leads to

〈∆shk〉 ≥ 0. (2.29)

2.2 Unified formulation based on reference probabil-

ities

In this section, we review a unified strategy to derive the nonequilibrium equalities intro-

duced in the previous section.

In Ref. [11], Crooks revealed that a unified approach to derive nonequilibrium equal-

ities is to compare the nonequilibrium process with the time-reversed process and obtain

a detailed fluctuation theorem, namely, the Crooks fluctuation theorem given by

P†[Γ†]P [Γ]

= e−β(W [Γ]−∆F ), (2.30)

where quantities with a superscript † are the ones in the time-reversed process. Later,

Hatano and Sasa derived the nonequilibrium equality (2.22) in steady states by comparing

the original dynamics with its “time-reversed dual” dynamics in a sense to be specified

later. In their derivation, they essentially used a detailed fluctuation theorem given by

P†[Γ†]P [Γ]

= e−∆φ[Γ]−∆sex[Γ]/kB , (2.31)

where quantities accompanied by † are, in this equality, the ones in the time-reversed dual

Table 2.1: Choices of the reference dynamics and the associated entropy productions.Specific choices of the reference dynamics lead to specific entropy productions with specificphysical meanings. The meaning of the “boundary term” is explained later.

Reference dynamics Entropy production σ

time reversal ∆sbath/kB + (boundary term)steady-flow reversal ∆shk/kB + (boundary term)

time reversal + steady-flow reversal ∆sex/kB + (boundary term)

19

process. Their work indicates that a further unification is possible, namely, a detailed

fluctuation theorem will be obtained when we compare the original dynamics with a

properly chosen reference dynamics. With this speculation, let us generalize the detailed

fluctuation theorems (2.30) and (2.31) to

Pr[Γ]

P [Γ]= e−σ[Γ], (2.32)

where Pr represents a reference probability of a reference path, and σ[Γ] is a formal

entropy production. In the case of the Crooks fluctuation theorem (2.12), the reference

is the time reversal, and σ[Γ] reduces to W [Γ] −∆F . In the case considered by Hatano

and Sasa, in a similar way, the reference is the time-reversed dual, and σ[Γ] reduces to

∆φ[Γ]+∆sex[Γ]/kB. Once we obtain the detailed fluctuation theorem (2.32), we succinctly

derive an integral nonequilibrium equality given by

〈e−σ〉 = 1, (2.33)

because

〈e−σ〉 =

∫DΓe−σP [Γ]

=

∫DΓPr[Γ]

= 1, (2.34)

where we use Eq. (2.32) to obtain the second line, and we use the normalization condition

for the reference probability to obtain the last line. In the same way, we obtain

〈Fe−σ〉 = 〈F〉r, (2.35)

where F [Γ] is an arbitrary functional, and 〈· · · 〉r represents the average over the reference

probability Pr[Γ].

In the rest of this section, we validate Eq. (2.32) in specific systems. To be precise,

we show that appropriate choices of the reference probability make the formal entropy

production σ reduce to physically meaningful entropy productions. The results are sum-

marized in Table 2.1.

2.2.1 Langevin system

First of all, we consider a one-dimensional overdamped Langevin system. One reason

why we deal with the Langevin system as a paradigm is that steady states can be simply

20

achieved by applying an external driving force. Moreover, the Langevin system is ther-

modynamically sound in that it relaxes to the thermal equilibrium state, i.e., the Gibbs

state after a sufficiently long time without the external driving force.

This part is mainly based on a review article by Seifert, i.e., Ref. [53].

Basic properties and definition of heat

Let x(t) denote the position of a particle at time t in a thermal environment with tem-

perature T . We consider a nonequilibrium process from time t = 0 to τ controlled by an

external parameter λ(t). The overdamped Langevin equation is given by

x(t) = µF (x(t), λ(t)) + ζ(t), (2.36)

where µ is the mobility, and F (x, λ) is a systematic force applied to the particle with the

position x when the external control parameter is λ, and ζ(t) represents a random force.

We assume that ζ(t) is a white Gaussian noise satisfying

〈ζ(t)ζ(t′)〉 = 2Dδ(t− t′), (2.37)

where D is the diffusion constant. The systematic force F (x, λ) consists of two parts, that

is,

F (x, λ) = −∂xV (x, λ) + f(x, λ). (2.38)

The first part is due to the conservative potential V (x, λ) and f(x, λ) is the external driv-

ing force. Under this Langevin dynamics, the probability to generate an entire trajectory

x starting from x0 = x(0) under a given entire protocol λ is calculated as

P [x|x0, λ] = N e−A[x,λ], (2.39)

where the action A[x, λ] of the trajectory x is

A[x, λ] =

∫ τ

0

dt

[(x− µF (x, λ))2

4D+ µ

∂xF (x, λ)

2

](2.40)

(see Appendix A for a derivation). Another strategy to describe the system is to trace

the probability p(x, t) to find the particle at x at time t. We can show p(x, t) to obey the

Fokker-Planck equation given by

∂tp(x, t) = −∂xj(x, t), (2.41)

21

where the probability current j(x, t) is defined as

j(x, t) = µF (x, λ(t))p(x, t)−D∂xp(x, t) (2.42)

(see Appendix A for a derivation). With this probability, the entropy of the system is

defined as the stochastic Shannon entropy of the system, i.e.,

s(t) = −kB ln p(x(t), t). (2.43)

The mean local velocity is defined as

v(x, t) =j(x, t)

p(x, t). (2.44)

When the external driving force is not applied, i.e., f(x, λ) = 0, the system relaxes to

the thermal equilibrium state given by

peq(x, λ) = e−β(V (x,λ)−F (λ)), (2.45)

where β is the inverse temperature and free energy F (λ) is defined as

e−βF (λ) =

∫dxe−βV (x,λ), (2.46)

because the kinetic energy contributes nothing due to the assumption of overdamping.

Next, we consider how we should define heat in this Langevin system [54]. The dissi-

pated heat is the energy flow from the system to the bath. This energy transfer is done by

the viscous friction force −γx and the thermal noise γζ(t), where γ = 1/µ is the friction

coefficient. Therefore, the “work” done by these force should be identified with the heat

flowing into the system. Thus we define the dissipated heat as

q[x] = −∫ τ

0

dtx(−γx+ γζ). (2.47)

Using the Langevin equation (2.36), we obtain

q[x] =

∫ τ

0

dtxF (x, λ). (2.48)

Now we can define the entropy production of the heat bath as

∆sbath =q

T. (2.49)

22

In this way, the concepts of the heat and entropy are generalized to the level of an

individual stochastic trajectory.

Steady-state properties and definitions of thermodynamic quantities

When the external driving force f(x, λ) is applied at a fixed λ, the system relaxes to a

nonequilibrium steady state pss(x, λ). In the analogy of the equilibrium state (2.45), let

us define a nonequilibrium potential φ(x, λ) by

pss(x, λ) = e−φ(x,λ). (2.50)

The steady current is defined as

jss(x, λ) = µF (x, λ)pss(x, λ)−D∂xpss(x, λ), (2.51)

and the mean velocity in the steady state is given by

vss(x, λ) =jss(x, λ)

pss(x, λ)

= µF (x, λ)−D∂x ln pss(x, λ)

= µF (x, λ) +D∂xφ(x, λ). (2.52)

Using this expression, we rewrite Eq. (2.48) as

q[x] =1

µ

∫ τ

0

dtxvss(x, λ)− kBT

∫ τ

0

dtx∂xφ(x, λ), (2.53)

where the Einstein relation D = µkBT is assumed. The first term is the inevitable

dissipation proportional to the steady mean velocity vss and is therefore identified as the

housekeeping heat, namely,

qhk[x] =1

µ

∫ τ

0

dtxvss(x, λ), (2.54)

and the second term is the excess contribution after the subtraction of the housekeeping

part from the total heat, and defined as

qex[x] = −kBT

∫ τ

0

dtx∂xφ(x, λ). (2.55)

23

Therefore, following Ref. [50], we define two kinds of entropy production as

∆shk[x] =kB

D

∫ τ

0

dtxvss(x, λ), (2.56)

and

∆sex[x] = −kB

∫ τ

0

dtx∂xφ(x, λ)

= kB

[−∆φ+

∫ τ

0

dtλ∂λφ(x, λ)

], (2.57)

because dφ = (∂xφ)dx + (∂λφ)dλ. Note that the ensemble average of the excess entropy

production vanishes in steady states because λ is independent of time, which justifies that

the excess entropy production is indeed an excess part due to non-adiabatic changes of

the control parameter λ.

Time reversal and ∆sbath

We consider a process starting from an initial probability distribution p0(x0) under a pro-

tocol λ, and the time-reversed process starting from an initial probability distribution

p†0(x†0) under the time-reversed protocol λ† defined by λ†(t) := λ(τ − t). Here, we do

not assume any relations between p0(x0) and p†0(x†0). Now we compare the realization

probability of the original path x and that of the time-reversed path x† defined by

x†(t) = x(τ − t). The original probability is

P [x|λ] = P [x|x0, λ]p0(x0)

= Np0(x0)e−A[x,λ], (2.58)

and the time-reversed probability is

P [x†|λ†] = P [x†|x†0, λ†]p†0(x†0)

= Np†0(x†0)e−A[x†,λ†]. (2.59)

Therefore, the formal entropy production defined in Eq. (2.32) reduces to

σ[x] = − lnP [x†|λ†]P [x|λ]

= (A[x†, λ†]−A[x, λ])− lnp†0(x†0)

p0(x0). (2.60)

24

The first term is called the bulk term, and the second term is called the boundary term,

because the second one arises from the boundary conditions p0 and p†0. Using Eq. (2.40),

we obtain

A[x†, λ†] =

∫ τ

0

dt

[(x†(t)− µF (x†(t), λ†(t)))2

4D+ µ

∂xF (x†(t), λ†(t))

2

]=

∫ τ

0

dt

[(−x(τ − t)− µF (x(τ − t), λ(τ − t)))2

4D+ µ

∂xF (x(τ − t), λ(τ − t))2

]=

∫ τ

0

dt

[(x(t) + µF (x(t), λ(t)))2

4D+ µ

∂xF (x(t), λ(t))

2

], (2.61)

where we change the integration variable from t to τ− t to obtain the last line. Therefore,

the bulk term is

A[x†, λ†]−A[x, λ] =1

kBT

∫ τ

0

dtxF (x, λ) =q[x]kBT

, (2.62)

where we use the definition (2.48) to obtain the last equality. By Eq. (2.49), we obtain

σ[x] =∆sbath

kB

− lnp†0(x†0)

p0(x0). (2.63)

Here, we assume that the initial state is in equilibrium

p0(x0) = peq(x0, λ0) = e−β(U(x0,λ0)−F (λ0)). (2.64)

Although the initial state of the reversed dynamics can be set to an arbitrary probability

distribution, we set it also to the canonical ensemble as

p†0(x†0) = peq(x†0, λ†0) = e−β(U(x†0,λ

†0)−F (λ†0))

= e−β(U(xτ ,λτ )−F (λτ )). (2.65)

In this case, we obtain

σ[x] =∆sbath

kB

+ β(∆U −∆F )

= β(q + ∆U −∆F )

= β(W −∆F ), (2.66)

which means that we obtain the Crooks fluctuation theorem (2.12). Therefore, we also

obtain the Jarzynski equality (2.4) succinctly.

Next, we assume nothing about p0, and set the initial probability of the reversed

25

dynamics to the final probability of the original dynamics, namely,

p†0(x†0) = p(x†0, τ) = p(xτ , τ). (2.67)

In this case, the boundary term reduces to the Shannon entropy production of the system

as

σ[x] =∆sbath

kB

− ln p(xτ , τ) + ln p(x0, 0)

=∆sbath + s(τ)− s(0)

kB

=∆sbath + ∆s

kB

. (2.68)

Therefore, we obtain

σ[x] =∆stot

kB

, (2.69)

and the detailed Seifert relation (2.27), which automatically derives the integral Seifert

relation (2.25).

Finally, we assume nothing about p0, and set the initial probability of the reversed

dynamics to p0. Then, we obtain

σ[x] = Ω[x], (2.70)

where

Ω[x] =∆sbath

kB

− ln p0(xτ ) + ln p0(x0) (2.71)

is the dissipation functional defined in Ref. [5] and used in the transient fluctuation the-

orem (2.2).

Dual (steady-flow reversal) and ∆shk

Next, we consider the steady-flow-reversed dynamics. To this aim, we define the dual of

the external force F (x, λ) as

F †(x, λ) = F (x, λ)− 2vss,F (x, λ)

µ, (2.72)

where we use vss,F to represent the steady velocity under force F . Let us demonstrate

that the steady state under force F is also the steady state under force F †. Comparing

26

Eq. (2.52), we obtain

µF †(x, λ) = −µF (x, λ)− 2D∂xφF (x, λ), (2.73)

or

µF †(x, λ) +D∂xφF (x, λ) = −(µF (x, λ) +D∂xφF (x, λ)). (2.74)

Since pss,F (x, λ) = e−φF (x,λ) is the steady solution of the Fokker-Planck equation (2.41),

we have

0 = ∂xjss,F (x, λ) = ∂x(µF (x, λ)pss,F (x, λ)−D∂xpss,F (x, λ))

= ∂x[(µF (x, λ) +D∂xφF (x, λ))pss,F (x, λ)]. (2.75)

Using Eq. (2.74), we obtain

0 = ∂x[(µF†(x, λ) +D∂xφF (x, λ))pss,F (x, λ)]

= ∂x(µF†(x, λ)pss,F (x, λ)−D∂xpss,F (x, λ)), (2.76)

which means pss,F is also the steady state solution under force F †, that is,

pss,F (x, λ) = pss,F †(x, λ) (=: pss(x, λ)), (2.77)

φF (x, λ) = φF †(x, λ) (=: φ(x, λ)). (2.78)

Comparing Eqs. (2.51) and (2.74), we have

jss,F (x, λ) = −jss,F †(x, λ) (=: jss(x, λ)). (2.79)

Therefore, the dual dynamics has the same steady state as the original dynamics and the

negative of the steady current in the original dynamics.

Now, we consider a process starting from an initial probability distribution p0(x0)

and the dual process starting from an initial probability distribution p†0(x0). The original

probability is

PF [x|λ] = PF [x|x0, λ]p0(x0)

= Np0(x0)e−AF [x,λ] (2.80)

27

and the dual probability is

PF † [x|λ] = PF † [x|x0, λ]p0(x0)

= Np†0(x0)e−AF† [x,λ]. (2.81)

Therefore, the formal entropy production reduces to

σ[x] = (AF † [x, λ]−AF [x, λ])− lnp†0(x0)

p0(x0). (2.82)

The bulk term can be calculated as

AF † [x, λ]−AF [x, λ]

=

∫ τ

0

dt

[(x− µF †)2

4D+ µ

∂xF†

2

]−∫ τ

0

dt

[(x− µF )2

4D+ µ

∂xF

2

]=

∫ τ

0

dt[ µ

4D(2x− µF − µF †)(F − F †) +

µ

2∂x(F

† − F )]. (2.83)

Using Eqs. (2.72) and (2.73), we obtain

AF † [x, λ]−AF [x, λ] =

∫ τ

0

dt

[1

D(x+D∂xφ)vss − ∂xvss

]=

1

D

∫ τ

0

dtxvss −∫ τ

0

dteφ∂xjss

=1

D

∫ τ

0

dtxvss, (2.84)

where we use jss = vsse−φ to obtain the second line and the fact that the divergence of

the current vanishes in the steady state to obtain the last line. By the definition (2.56),

we obtain

σ[x] =∆shk

kB

− lnp†0(x0)

p0(x0). (2.85)

When we set p†0 to p0, the formal entropy production reduces to

σ[x] =∆shk

kB

, (2.86)

and therefore we obtain the integral fluctuation theorem for the housekeeping entropy

production (2.28).

28

Time-reversed dual and ∆sex

We consider a process starting from an initial probability distribution p0(x0) under a

protocol λ, and compare it with the dual process starting from an initial probability

distribution p†0(x†0) under the time-reversed protocol λ†. The original probability is

PF [x|λ] = PF [x|x0, λ]p0(x0)

= Np0(x0)e−AF [x,λ] (2.87)

and the time-reversed dual probability is

PF † [x†|λ†] = PF † [x†|x†0, λ†]p†0(x†0)

= Np†0(x†0)e−AF† [x†,λ†] (2.88)


σ[x] = (AF † [x†, λ†]−AF [x, λ])− lnp†0(x†0)

p0(x0). (2.89)

The bulk term is calculated as

AF † [x†, λ†]−AF [x, λ]

=

∫ τ

0

dt

[(−x− µF †)2

4D+ µ

∂xF†

2

]−∫ τ

0

dt

[(x− µF )2

4D+ µ

∂xF

2

]=

∫ τ

0

dt[− µ

4D(F + F †)(−2x+ µ(F − F †)) +

µ

2∂x(F

† − F )]. (2.90)

Using Eqs. (2.72) and (2.73), we obtain

AF † [x†, λ†]−AF [x, λ] =

∫ τ

0

dt [(∂xφ)(−x+ vss)− ∂xvss]

= −∫ τ

0

dtx∂xφ−∫ τ

0

dteφ∂xjss

= −∫ τ

0

dtx∂xφ. (2.91)

By the definition (2.57), we obtain

σ[x] =∆sex

kB

− lnp†0(x†0)

p0(x0). (2.92)

29

We assume that the initial state is the nonequilibrium steady state given by

p0(x0) = e−φ(x0,λ0) (2.93)

and set the initial state of the time-reversed dual dynamics to the nonequilibrium steady

state as

p†0(x†0) = e−φ(x†0,λ†0)

= e−φ(xτ ,λτ ). (2.94)

Then, we obtain

σ[x] =∆sex

kB

+ φ(xτ , λτ )− φ(x0, λ0)

=∆sex

kB

+ ∆φ. (2.95)

Therefore, we reproduce the Hatano-Sasa relation (2.22) in a simple manner.

2.2.2 Hamiltonian system

Next, we consider a Hamiltonian system consisting of a system and a heat bath with

inverse temperature β. We consider only time reversal as the reference dynamics because

it is difficult to define steady flow in a Hamiltonian system in general.

This part is partly based on Ref. [55].

Setup

Let z denote the position in the phase space of the total system. We separate the degrees

of freedom z into two parts as z = (x, y), where x denotes the degrees of freedom of the

system, and y is the degrees of freedom of the bath. We assume that the Hamiltonian of

the total system can be decomposed into

Htot(z, λ) = H(x, λ) +Hint(x, y, λ) +Hbath(y), (2.96)

where λ is an external control parameter. We also assume that the Hamiltonian is invari-

ant under time reversal. We vary λ from time t = 0 to τ , and the system is subject to a

nonequilibrium process. The free energy of the bath Fbath is defined by

e−βFbath =

∫dye−βHbath(y), (2.97)

30

and is invariant during the process. Moreover, we define an effective Hamiltonian of the

system Hef(x, λ) by tracing out the degrees of freedom of the bath as

e−βHef(x,λ)e−βFbath = e−βH(x,λ)

∫dye−β(Hint(x,y)+Hbath(y)). (2.98)

The free energy of the total system Ftot(λ) defined by

e−βFtot(λ) =

∫dze−βHtot(z,λ) (2.99)

and the free energy F (λ) based on Hef(x, λ) defined by

e−βF (λ) =

∫dxe−βHef(x,λ) (2.100)

are related by

Ftot(λ) = F (λ) + Fbath. (2.101)

Time reversal

We compare the original process z starting from an initial probability distribution of the

total system p0(z0) with the time-reversed process z† starting from an initial probability

distribution of the total system p†0(z†0), where z†(t) = z∗(τ − t) and the superscript ∗represents the sign reversal of momenta. We note that the probability to realize a path

z is the same as the probability to have z0 in the initial state because the Hamiltonian

system is deterministic as a whole. Therefore, we obtain

σ[z] = − lnP [z†|λ†]P [z|λ]

= − lnp†0(z†0)

p0(z0). (2.102)

Now, we assume that the initial state is the equilibrium state of the total Hamiltonian

p0(z0) = e−β(Htot(z0,λ0)−Ftot(λ0)). (2.103)

Moreover, we set the initial state of the reversed process to the equilibrium state of the

total Hamiltonian

p†0(z†0) = e−β(Htot(zt,λt)−Ftot(λt)). (2.104)

31

Thus, we obtain

σ[x] = β(∆Htot −∆Ftot). (2.105)

Since the total system is isolated, the difference of the total Hamiltonian is due to the

work done by the external controller, that is,

∆Htot = W. (2.106)

Therefore, noting that Fbath is independent of λ, we obtain

σ = β(W −∆F ), (2.107)

which gives the Crooks fluctuation theorem (2.12) and the Jarzynski equality (2.4).

Next, we assume that the initial state of the system is not correlated with the initial

state of the bath and that the initial state of the bath is the canonical ensemble, namely,

p0(z0) = p0(x0)e−β(Hbath(y0)−Fbath). (2.108)

In addition, we set the initial state of the time-reversed dynamics to the product state of

the final probability distribution of the system in the original process and the canonical

ensemble of the bath

p†0(z†0) = pτ (x†0)e−β(Hbath(y†0)−Fbath)

= pτ (xτ )e−β(Hbath(yτ )−Fbath). (2.109)


σ[x] = β∆Hbath − ln pτ (xτ ) + ln p0(x0). (2.110)

We define an unaveraged Shannon entropy of the system by s(t) = −kB ln pt(xt), and we

have

σ = β∆Hbath +∆s

kB

. (2.111)

The energy change of the bath ∆Hbath can be regarded as the heat Q dissipated from

the system to the bath. Moreover, the heat Q is related to the entropy production of the

32

bath ∆sbath as Q/T = ∆sbath. Thus, we obtain

σ =∆sbath + ∆s

kB

=∆stot

kB

, (2.112)

where ∆stot = ∆sbath + ∆s is the total entropy production. Therefore, we automatically

reproduce the detailed Seifert relation (2.27) and the integral Seifert relation (2.25).

The results obtained in this section are summarized in Table 2.2.

Table 2.2: Summary of choices of the reference probability and specific meanings of theentropy production.

Reference dynamics Reference initial state p†0(x(†)0 ) Entropy production σ

time reversal canonical peq(xτ , λτ ) dissipated work β(W −∆F )

time reversal final state pτ (xτ ) total ∆stot/kB

time reversal initial state p0(xτ ) dissipation functional Ω

dual initial state p0(x0) housekeeping ∆shk/kB

time-reversed dual steady state pss(xτ , λτ ) excess ∆φ+ ∆sex/kB

33

Chapter 3

Review of Information

Thermodynamics

In this chapter, we review thermodynamics with measurements and feedback control.

Historically, Maxwell pointed out that thermodynamics, specifically the second law of

thermodynamics, should break down when an intelligent being, known as Maxwell’s de-

mon, controls the system by utilizing information obtained by measurements. Since then,

numerous researches have been done on the foundation of the second law of thermody-

namics [56], and thermodynamics of information processing is established [57]. Since the

nonequilibrium equalities reviewed in the previous chapter are generalizations of the sec-

ond law of thermodynamics, the second-law-like inequality of information processing can

be extended to nonequilibrium equalities.

First, we trace historical discussions on Maxwell’s demon. Then, we formulate the

second law of information thermodynamics from a modern point of view. Next, nonequi-

librium equalities of information thermodynamics are reviewed. Finally, we review exper-

imental demonstrations of Maxwell’s demon.

3.1 Maxwell’s demon

In this section, we review historical discussions on Maxwell’s demon.

3.1.1 Original Maxwell’s demon

Maxwell’s demon was proposed in his book titled “Theory of Heat” published in 1871

[12]. In the second last section of the book, Maxwell discussed the “limitation of the

second law of thermodynamics” in an example with an intelligent being, which was later

christened Maxwell’s demon by Lord Kelvin.

34

(a)

(b)

Figure 3.1: Schematic illustration of the gedankenexperiment of Maxwell’s demon. (a)Second law. Initially, the left box is filled with blue (slower and colder) molecules, andthe right box is filled with red (faster and hotter) molecules. When the window of thedivision is open, the entire system becomes uniform in temperature. (b) Function ofMaxwell’s demon. Initially, the temperature is uniform. The demon measures the velocityof molecules and let the red particles go through the window from the left box to the rightbox and let the blue particles go from the right to the left. Nonuniformity of temperatureis then achieved without work or heat exchange.

Let us consider a vessel filled with gas molecules. The vessel is divided into two parts,

and the division has a small window, through which a molecule passes from one side to

the other when the window is open. When the window opens and the temperature of

one side is different from that of the other side, the second law of thermodynamics states

that the temperature becomes uniform (see Fig. 3.1 (a)). Maxwell’s demon achieves the

reverse process of this phenomenon (see Fig. 3.1 (b)). At an initial time, the temperature

is uniform throughout the vessel. The demon observes molecules in the vessel. Some

molecules are faster than the mean velocity and others are slower because of thermal

fluctuations. The demon opens and closes the window to allow only the faster-than-

average molecules to pass from the left side to the right side, and only the slower-than-

average molecules to pass from the right to the left. After some time, the demon succeeds

in raising the temperature of the right side and lowering the temperature of the left without

the expenditure of work. This means that the entropy is reduced in an isolated system,

which apparently contradicts the second law of thermodynamics. In summary, Maxwell

demonstrated that the control of the system based on the outcomes of the measurement

can reduce the entropy of the system beyond the restriction from the second law of

thermodynamics.

35

Figure 3.2: Protocol of the Szilard engine. A single-particle gas particle is enclosed inthe box, which is surrounded by a heat bath at temperature T . A partition is insertedin the middle, and the position of the particle is measured. Based on the measurementoutcome, we decide in which direction we shift the division. We isothermally expand thedivision to the end and remove the division.

3.1.2 Szilard engine

In 1929, a simplest model of Maxwell’s demon, now known as the Szilard engine, was pro-

posed [13]. Although the Szilard engine is apparently different from the original Maxwell’s

gedankenexperiment, it captures the essential features of the demon of utilizing measure-

ment and feedback control to reduce the entropy of a system. Moreover, the Szilard engine

enables us to quantitatively analyze the role of the information.

We elaborate on the protocol of the Szilard engine (see Fig. 3.2). An ideal classical

gas molecule is confined in a box with volume V , and the box is surrounded by an

isothermal environment with temperature T . We insert a division in the middle of the

box and separate the box into two parts with the same volume V/2. Then, we measure

the position of the particle to determine whether the particle is in the left or right part.

We assume this measurement is error-free. When we find the particle is in the left, we

isothermally shift the division to the right end. On the other hand, when we find the

particle is in the right, we isothermally shift the division to the left end. In both cases,

we can extract a positive work of kBT ln 2 (see the discussion below for the derivation of

this result) from the particle in these processes. We remove the division and the system

returns to its initial state. Therefore, we can repeatedly extract work from this isothermal

cycle.

Szilard pointed out that the correlation made by the measurement is the resource for

36

the entropy reduction and work extraction. The measurement process creates a correlation

between the position of the particle x and the measurement outcome y. Let us set the

origin of x at the middle. When x > 0 (x < 0), we obtain y = R (y = L), where R (L)

means the right (left) . After the process of feedback control, namely isothermal shifting,

this correlation vanishes because the particle can now be present in the entire system,

so x can be positive or negative regardless of the value of y. Therefore, in the feedback

process, we extract a positive work at the cost of eliminating the correlation between x

and y.

Let us quantitatively analyze this protocol from a modern point of view. By the

position measurement, we obtain the Shannon information

I = ln 2. (3.1)

In the process of isothermal expansion, we extract work Wext. Because we assume that

the gas is ideal, the equation of state reads pV = kBT . Therefore, the work is calculated

as

Wext =

∫ V

V/2

pdV = kBT

∫ V

V/2

dV

V= kBT ln 2 = kBTI. (3.2)

Therefore, we conjecture that the information obtained by the measurement can be quan-

titatively converted to the work during the feedback process.

As explained above, Szilard revealed that we can utilize the correlation established

by the measurement to reduce the entropy of the system. He also pointed out that the

entropy reduction achieved in the feedback process must be compensated by a positive

entropy production during the process to establish the correlation to be consistent with the

second law of thermodynamics of the entire process. However, it remained unexplained

why the measurement process should be accompanied by a positive entropy production.

3.1.3 Brillouin’s argument

An answer to the question was presented by Brillouin in 1951 [14]. He argued, in the

original setup of Maxwell’s demon, that the demon creates more entropy when it observes

molecules than the entropy reduction due to his feedback control. Therefore, the obser-

vation process compensates the entropy reduction of the gas, and as a result the entire

process is consistent with the second law of thermodynamics.

Brillouin assumed that, when the demon observes molecules, the demon needs to shed

a probe light to molecules. However, the demon and the system are surrounded by an

environment at temperature T with the blackbody radiation. Therefore, the energy of

37

the probe photon should be sufficiently larger than the thermal energy kBT to distinguish

the probe from background noises. Thus, the frequency of the probe photon ν satisfies

hν kBT, (3.3)

where h is the Planck constant. The demon observes a molecule by absorbing a photon

scattered by the molecule. Therefore, the entropy production of the demon by the single

observation is given by

∆Sdemon =hν

T kB. (3.4)

Let TL (TR) represent the temperature of the left (right) side satisfying

TL = T − 1

2∆T, TR = T +

1

2∆T, (3.5)

where ∆T is the temperature difference satisfying ∆T T . The demon transfers a fast

molecule in the left box with kinetic energy 32kBT (1 + ε1) to the right box, and does a

slow molecule in the right box with kinetic energy 32kBT (1 − ε2) to the left box, where

ε1, ε2 > 0, and ε1 and ε2 are of the order of one. As a result, heat

Q =3

2kBT (ε1 + ε2) (3.6)

is transferred from the left box to the right box, and the entropy reduction is bounded as

−∆Ssys ≤ Q

(1

TL

− 1

TR

)' Q

∆T

T 2=

3

2kB(ε1 + ε2)

∆T

T kB, (3.7)

because ∆T/T 1 and ε1 + ε2 ∼ 1. Therefore, comparing this equation with Eq. (3.4),

we conclude that the entropy production of the demon due to the measurement is far

beyond the entropy reduction achieved by the feedback control. Thus, the second law

remains valid for the entire system. A similar discussion can be also done for the Szilard

engine.

In this way, Brillouin argued that the entropy production for the measurement process

exceeds the entropy reduction during the feedback process, and therefore the total entropy

production of the system and the demon is positive. However, his analysis depends on

a specific model of the measurement using a photon as the probe, and the idea that the

work gain by the feedback control is compensated by the work cost of the measurement

is not always true.

38

0 1Figure 3.3: Structure of a 1-bit memory. A particle is confined in a bistable potential.In the zero (one) state, the particle is in the left (right) well.

3.1.4 Landauer’s principle

Landauer argued that the energy cost is needed not for measurement processes to obtain

information but for erasure processes of the obtained information from the memory [15].

He considered a 1-bit memory consisting of a particle in a bistable potential as shown in

Fig. 3.3. We label the particle in the left well as the zero state, and the particle in the

right well as the one state. In the erasure process, we restore the particle to the standard

state, namely the zero state. Before the erasure, we do not know whether the particle is

in the left or right well. Therefore, the process is a two-to-one mapping, and cannot be

realized by a deterministic frictionless protocol. Thus, a protocol must involve a process

with friction to erase the information. In this way, dissipation is inevitable to erase the

information stored in the memory.

The erasure process in which the system is brought to the zero state is logically

irreversible because we cannot recover the state before the process from the state after the

process. Landauer argued that logical irreversibility implies physical irreversibility, which

is accompanied by dissipation. Therefore, logical irreversible operations such as erasure

cause heat dissipation.

In a 1-bit symmetric memory, where the zero and one states have the same entropy,

the erasure from the randomly-distributed state to the standard state means the entropy

reduction by kB ln 2. To compensate this entropy reduction, heat dissipation must occur.

Therefore, to erase the information stored in a 1-bit memory, we have inevitable heat

dissipation of kBT ln 2. This is a famous rule known as Landauer’s principle.

In summary, Landauer argued that the cost for erasure of the stored information

compensates the gain in feedback processes. However, his argument is crucially dependent

on the structure of the symmetric memory, and does not apply to general cases. In fact,

39

erasure in an asymmetric memory provides a counter-example of Landauer’s principle

[57].

3.2 Second law of information thermodynamics

In this section, we review the second law of information thermodynamics from a modern

point of view. We restrict our attention to classical cases, because it is sufficient for the

aim of this thesis. First of all, we shortly introduce classical information quantities because

they are crucial ingredients of information thermodynamics. Then, the second law of a

system under feedback control is discussed. After that, the second law of a memory, which

is a thermodynamic model of the demon, is presented. Finally, we demonstrate that the

conventional second law is recovered for the entire system.

3.2.1 Classical information quantities

In this section, we introduce three classical information quantities: the Shannon entropy,

the Kullback-Leibler divergence, and the mutual information based on Refs. [57, 58].

Shannon entropy

First of all, we introduce the Shannon entropy. Let x denote a probability variable and Xdenote the sample space, namely x ∈ X . When X is a discrete set, we define the Shannon

entropy of a probability distribution p(x) as 1

SX [p] = −∑x∈X

p(x) ln p(x). (3.8)

On the other hand, when X is continuous, we would naively define the Shannon entropy

of a probability distribution density p(x) by

−∫Xp(x)dx ln(p(x)dx) = −

∫Xdx p(x) ln p(x)−

∫Xdx p(x) ln(dx). (3.9)

However, the second term is divergent in the limit of dx → 0. Therefore, we define the

Shannon entropy as

SX [p] = −∫Xdx p(x) ln p(x). (3.10)

1 In the context of quantum information theory, the symbol S usually denotes the von Neumannentropy instead of the Shannon entropy. However, in this thesis, we denote the Shannon entropy by Sbecause it is the convention in the field of classical statistical mechanics. There would be no confusionbecause we do not use the von Neumann entropy in this thesis.

40

Although p(x)dx is invariant under a transformation of variable, p(x) alone is not invari-

ant. Therefore, the continuous Shannon entropy as defined in Eq. (3.10) is not invariant

under transformation of the coordinates.

The unaveraged Shannon entropy

s(x) = − ln p(x) (3.11)

is an indicator of how rare an event x is. In fact, s(x) increases as p(x) decreases.

The Shannon entropy is the ensemble average of this rarity. The reason why we use the

logarithm is to guarantee the additivity of the Shannon entropy when we have independent

events. Let us assume X = X1 × X2 and x = (x1, x2). Then, when p(x) = p1(x1)p2(x2),

we have

SX (p) = SX1(p1) + SX2(p2). (3.12)

Here, we demonstrate that the Shannon entropy is invariant under a Hamiltonian

dynamics. In this case, X is phase space, and the dynamics is deterministic. Let xi (xf)

denote the initial (final) position in X and pi (pf) denote the initial (final) probability

distribution. Since the probability is conserved, we have

pi(xi)dxi = pf(xf)dxf . (3.13)

In addition, Liouville’s theorem states that

dxi = dxf , (3.14)

which, together with Eq. (3.13), leads to

pi(xi) = pf(xf). (3.15)

Therefore, the initial Shannon entropy

SXi = −∫Xdxi p(xi) ln p(xi) (3.16)

has the same value as the final Shannon entropy

SXf = −∫Xdxf p(xf) ln p(xf), (3.17)

41

namely,

SXi = SXf . (3.18)

Thus, the continuous Shannon (3.10) entropy is invariant in time under the Hamiltonian

dynamics.

Kullback-Leibler divergence

Next, we introduce the Kullback-Leibler divergence or the relative entropy. This quantity

is defined as a relative logarithmic distance between two probability distributions p and

q on the same sample space X . When X is discrete, we define the Kullback-Leibler

divergence as

SX [p||q] = −∑x∈X

p(x) lnq(x)

p(x). (3.19)

On the other hand, when X is continuous, we define the Kullback-Leibler divergence as

SX [p||q] = −∫Xdx p(x) ln

q(x)

p(x). (3.20)

We note that the continuous Kullback-Leibler divergence is invariant under a transforma-

tion of the coordinates.

Using an inequality

lnq(x)

p(x)≤ q(x)

p(x)− 1, (3.21)

we obtain

SX [p||q] ≥ −∫Xdx p(x)

[q(x)

p(x)− 1

]= −

∫Xdx q(x) +

∫Xdx p(x)

= 0, (3.22)

where the equality is achieved if and only if p(x) = q(x) (p-almost everywhere). There-

fore, the Kullback-Leibler divergence is a kind of distance to measure how different two

probabilities are.

42

SX SY

SXY

IX :Y

Figure 3.4: Schematic illustration of the definition of the mutual information. The twocircles represent degrees of uncertainty of X and Y . The union of the circles represents adegree of uncertainty of X ×Y . Therefore, the mutual information defined by Eq. (3.23)corresponds to the overlap of the degrees of uncertainty of X and Y , which describescorrelations between X and Y .

Mutual information

We consider the mutual information between two sample spaces X and Y . Let pX×Y(x, y)

denote a joint probability distribution of (x, y) ∈ X × Y . The marginal probability

distributions are defined as pX (x) =∫Y dy p

X×Y(x, y) and pY(y) =∫X dx p

X×Y(x, y). We

define the mutual information as

IX :Y = SX [pX ] + SY [pY ]− SX×Y [pX×Y ], (3.23)

which represents the overlap of uncertainty of X and Y (see Fig. 3.4). If the systems

are independent of each other, i.e., p(x, y) = pX (x)pY(y), we obtain I = 0 due to the

additivity of the Shannon entropy. Moreover, we represent the mutual information in

terms of the Kullback-Leibler divergence as

IX :Y = SX×Y [pX×Y ||pXpY ]. (3.24)

Therefore, the mutual information quantifies how different pX×Y is from the non-correlated

probability distribution pXpY , that is, how correlated X and Y are. Since the Kullback-

Leibler divergence is not negative, we obtain

IX :Y ≥ 0. (3.25)

43

system outcomeZ M

tm zm

H(z,)

m

H(z,m)

tf zf

ti zi

time

measurement

feedback control

Figure 3.5: Flow chart of feedback control. From time ti to tm, the system evolves intime by a Hamiltonian H(z, λ). At time tm, we perform a measurement on the systemand obtain an outcome m. We conduct feedback control from time tm to tf . In otherwords, the time evolution after the measurement is governed by a Hamiltonian H(z, λm)that depends on the measurement outcome m.

The explicit form of the mutual information is

IX :Y = −∫Xdx

∫Ydy pX×Y(x, y) ln

pX (x)pY(y)

pX×Y(x, y). (3.26)

Therefore, we define the unaveraged mutual information as

i(x, y) = − lnpX (x)pY(y)

pX×Y(x, y)(3.27)

or

i(x, y) = − ln pX (x) + ln pX|Y(x|y), (3.28)

where pX|Y(x|y) is the probability distribution function of x conditioned by y. Therefore,

the unaveraged mutual information quantifies the decrease of the unaveraged Shannon

entropy of the system X due to the fact that we know the system Y is in a state y.

3.2.2 Second law under feedback control

In this section, we formulate the second law of a system under feedback control. The

mutual information plays a crucial role in quantifying the gain of feedback control.

44

Setup

We consider feedback control on a Hamiltonian system with phase space Z (see Fig. 3.5).

At initial time ti, the system is at position zi sampled from an initial probability distri-

bution pi(zi). From time ti to tm, the system evolves under a Hamiltonian H(z, λ), and

ends up with a point zm with a probability distribution pm(zm). At time tm, we measure

quantities of the system (e.g. positions, velocities and the number of particles in a given

region) and obtain a measurement outcome m. Let M denote the sample space of the

outcome. The detail of the measurement is modeled by conditional probabilities

p(m|zm), (3.29)

which is the probability to obtain the outcome m under the condition that the system is

at the position zm at time tm. Now that we have the outcome m, we know more detailed

information on the system than we did before the measurement. In fact, the probability

distribution of zm under the condition that we have obtained m is calculated by the Bayes

theorem as

pm(zm|m) =p(m|zm)pm(zm)

p(m), (3.30)

where the probability distribution of the outcome m is defined by

p(m) =

∫dzm p(m|zm)pm(zm). (3.31)

From time tm to tf , the system evolves under a Hamiltonian H(z, λm). Here, we conduct

feedback control of the system, that is, adjust the Hamiltonian in accordance with m via

the control parameter λm. Therefore, the protocol after the measurement is conditioned

by m. Let zf denote the final position and pf(zf |m) be the final probability distribution

conditioned by m. The unconditioned final probability distribution is calculated as

pf(zf) =∑m∈M

pf(zf |m)p(m). (3.32)

Shannon entropy production

We evaluate the Shannon entropy production in the above-described setup. Since Hamil-

tonian dynamics does not change the Shannon entropy (see Eq. (3.18)), we obtain

SZ [pi] = SZ [pm]. (3.33)

45

By the same reason, we also obtain

SZ [pm(·|m)] = SZ [pf(·|m)], ∀m ∈M. (3.34)

To evaluate entropy production of the system, let us compare the averaged final entropy

SZf =∑m∈M

p(m)SZ [pf(·|m)] (3.35)

with the initial entropy

SZi = SZ [pi]. (3.36)

By Eq. (3.34), the final entropy is calculated as

SZf =∑m∈M

p(m)SZ [pm(·|m)]

= −∫Zdzm

∑m∈M

p(m)pm(zm|m) ln pm(zm|m)

= −∫Zdzm

∑m∈M

pm(zm,m) ln pm(zm|m). (3.37)

On the other hand, by Eq. (3.33), the initial entropy can be transformed as

SZi = SZ [pm] = −∫Zdzm pm(zm) ln pm(zm)

= −∫Zdzm

∑m∈M

pm(zm,m) ln pm(zm). (3.38)


SZf − SZi = −∫Zdzm

∑m∈M

i(zm,m) = −IZ:M. (3.39)

Therefore, the Shannon entropy production of the system is the negative of the mutual

information obtained by the measurement.

Second law under feedback control

We separate the degrees of freedom z into that of the system x and that of the heat bath

b as z = (x, b). Let X and B denote phase spaces of the system and the bath, respectively.

46

We assume that the total Hamiltonian reads

HZ(z, λ) = HX (x, λ) +HB(b) +H int(z, λ), (3.40)

where the last term on the right-hand side is the interaction Hamiltonian and is assumed

to vanish at the initial and final times.

First, we assume that the initial state is the product state of an initial probability

distribution of the system and the canonical ensemble of the heat bath as

pZi (zi) = pXi (xi)pBeq(bi), (3.41)

where we define

pBeq(b) = e−β(HB(b)−FB) (3.42)

and

e−βFB =

∫db e−βH

B(b). (3.43)

In this case, since xi and bi are not correlated, the initial total Shannon entropy is calcu-

lated as

SZi = SXi + β(〈HB〉i − FB), (3.44)

where the initial energy of the bath is defined as

〈HB〉i =

∫Bdbi H

B(bi)pBeq(bi). (3.45)

Using the final probability distribution of the total system pZf (xf , bf), we define the

marginal final probability as

pXf (xf) =

∫Bdbf p

Zf (xf , bf). (3.46)

We calculate the relative entropy between the final state of the total system and the

product state of the final state of the system and the canonical state of the heat bath as

SZ [pZf ||pXf pBeq] = −∫Xdxf

∫Bdbf p

Zf (xf , bf) ln pXf (xf)e

−β(HB(bf)−FB) − SZf= SXf + β(〈HB〉f − FB)− SZf . (3.47)

47

Since the relative entropy is positive, we obtain

SZf ≤ SXf + β(〈HB〉f − FB). (3.48)

Therefore, comparing Eq. (3.39), we obtain

∆SX + βQ ≥ −IZ:M, (3.49)

where we identify the dissipated heat with the energy difference of the bath as

Q = 〈HB〉f − 〈HB〉i. (3.50)

Moreover, since the measurement outcome m should depend only on the degrees of free-

dom of the system xm, we have

p(m|zm) = p(m|xm), (3.51)

and therefore

i(zm,m) = − ln pZ(zm) + ln p(zm|m)

= − ln pM(m) + ln p(m|zm)

= − ln pM(m) + ln p(m|xm)

= i(xm,m). (3.52)

Thus, we obtain

∆SX + βQ ≥ −IX :M. (3.53)

The left-hand side of Eq. (3.53) means the sum of the Shannon entropy production of the

system and the entropy production of the heat bath. Therefore, Eq. (3.53) demonstrates

that the total entropy production can be reduced by the feedback control by the amount

of the mutual information IX :M.

To proceed further, we assume that the initial state of the system is in equilibrium

with the inverse temperature β given by

pXi (xi) = pXeq(xi, λi), (3.54)

48

where we define the canonical ensemble as

pXeq(x, λ) = e−β(HX (x,λ)−FX (λ)) (3.55)

and the free energy as

e−βFX (λ) =

∫Xdx e−βH

X (x,λ). (3.56)

The initial Shannon entropy is

SXi = −∫Xdxi p

Xi (xi) ln pXi (xi)

= β(〈HX 〉i − FXi ), (3.57)

where we define the initial energy of the system as

〈HX 〉i =

∫Xdxi p

Xi (xi)H

X (xi, λi). (3.58)

On the other hand, since the relative entropy is positive, we obtain

SX [pXf (·|m)||pXeq(·, λm,f)] ≥ 0. (3.59)

This relation can be rewritten as

β

∫Xdxf p

Xf (xf |m)(HX (xf , λm,f)− FX (λm,f))− SX [pXf (·|m)] ≥ 0. (3.60)

Averaging this over m with the probability p(m), we obtain

β(〈HX 〉f − FXf ) ≥ SXf , (3.61)

where we define the final internal energy as

〈HX 〉f =

∫Xdxf

∑m∈M

pXf (xf ,m)HX (xf , λXm,f) (3.62)

and the final free energy as

FXf =∑m∈M

p(m)FX (λm,f). (3.63)

49

Therefore, Eq. (3.53) reduces to

β(〈∆HX 〉 −∆FX +Q) ≥ −IX :M. (3.64)

Identifying the energy change of the total system 〈∆HX 〉+Q with the work W performed

on the system, we obtain

W −∆FX ≥ −kBTIX :M. (3.65)

We can rewrite this equation as

−W ≤ −∆FX + kBTIX :M, (3.66)

which means that we can extract more work from the system than the conventional second

law of thermodynamics by the amount of the mutual information IX :M obtained by the

measurement. This form of the second law under feedback control was first formulated in

Ref. [18] in a quantum system.

In summary, we formulate the second law under feedback control, and reveal that the

gain of feedback control is precisely quantified by the mutual information obtained by the

measurement. This is a rigorous formulation of Szilard’s idea that the correlation made

by the measurement can be utilized as a resource for the entropy reduction.

3.2.3 Second laws of memories

In this section, we formulate second laws of memories during a measurement process

and during an erasure process. The second laws of memories were discussed in quantum

systems in Ref. [19]. Here, we consider a classical version of this study.

Measurement process

First, we consider a measurement process. The phase space of the total system is denoted

by Z and the system is subject to a Hamiltonian dynamics. We assume that the total

system consists of three parts: the system, the memory, and a heat bath with phase spaces

X , Y , and B, respectively. Phase-space positions in Z, X , Y , and B are denoted by z, x,

y, and b, respectively. Moreover, we assume that the sample space of the memory Y is

the disjoint union of Ym (m ∈M = 0, · · · ,M). An outcome m is stored when y ∈ Ym.

Therefore, M can be regarded as a coarse-grained sample space of Y . We assume that

50

the total Hamiltonian is decomposed into

HZ(x, y, b, λ) = HX (x, λ) +HY(y, λ) +HB(b) +H int(x, y, b, λ), (3.67)

where the last term on the right-hand side is the interaction term, which is assumed to

vanish at the initial and final times.

Since the total system is a Hamiltonian system, the Shannon entropy of the total

system is conserved:

SZi = SZf . (3.68)

We assume that the initial state is a product state as

pZi (zi) = pXi (xi)pYi (yi)p

Beq(bi), (3.69)

and then we obtain

SZi = SXi + SYi + β(〈HB〉i − FB). (3.70)

Because of the positivity of the relative entropy, we obtain

S[pZf ||pX×Yf pBeq] ≥ 0

⇔ SX×Yf + β(〈HB〉f − FB) ≥ SZf . (3.71)

Substituting Eqs. (3.68) and (3.70), we obtain

SX×Yf − SXi − SYi + βQ ≥ 0, (3.72)

where

Q = 〈HB〉f − 〈HB〉i (3.73)

Using the definition of the mutual information, we obtain

∆SX + ∆SY + βQ ≥ IX :Y . (3.74)

Since we perform feedback control based on m, we should evaluate the entropy production

51

by IX :M instead of IX :Y . The difference between these two values is calculated as

IX :Y − IX :M = −∫Xdxf

∫Ydyf p

X×Yf (xf , yf) ln

pXf (xf)pYf (yf)

pX×Yf (xf , yf)

+

∫Xdxf

∑m∈M

pX×Mf (xf ,m) lnpXf (xf)p

M(m)

pX×Mf (xf ,m), (3.75)

where the joint probability of X and M is defined as

pX×Mf (xf ,m) =

∫Ymdyf p

X×Yf (xf , yf). (3.76)


IX :Y − IX :M = −∫Xdxf

∑m∈M

∫Ymdyf p

X×Yf (xf , yf) ln

pYf (yf)pX|Mf (xf |m)

pX×Yf (xf , yf). (3.77)

Since the right-hand side is in the form of the relative entropy, we derive

IX :Y − IX :M ≥ 0. (3.78)

We note that this result is natural sinceM is the coarse-grained sample space of Y . Thus,

Eq. (3.74) reduces to

∆SX + ∆SY + βQ ≥ IX :M. (3.79)

We assume that, before the measurement process, the system is in equilibrium and

the memory is prepared to be a fixed standard state pYst(yi), namely, a local equilibrium

state in Y0 given by

pYst(yi) = pY0eq (yi, λi), (3.80)

where we define

pYmeq (y, λ) = χYm(y)e−β(HY (y,λ)−FYm (λ)), (3.81)

and χYm(y) is the characteristic function of a region Ym; the conditional free energy is

defined by

e−βFYm (λ) =

∫Ymdy e−βH

Y (y,λ). (3.82)

52

The initial entropy of the memory is

SYi = β(〈HY〉i − FYi ). (3.83)

Using the positivity of the relative entropy, we obtain

SY [pYf ||pYref ] ≥ 0, (3.84)

where we define a reference probability by

pYref(yf) =∑m∈M

p(m)pYmeq (yf , λm,f). (3.85)


−∫Ydyf p

Yf (yf) ln

∑m∈M

p(m)pYmeq (yf , λm,f) ≥ SYf

⇔ −∑m∈M

∫Ymdyf p

Yf (yf) ln p(m)pYmeq (yf , λm,f) ≥ SYf

⇔ SM + β(〈HY〉f − FYf ) ≥ SYf , (3.86)

where we use the fact that pYmeq = 0 outside Ym, and the final free energy is defined with

FYf =∑m∈M

p(m)FYm(λm,f). (3.87)

Therefore, the entropy production of the memory is bounded as

∆SY ≤ SM + β(∆〈HY〉 −∆FY). (3.88)

On the other hand, we can derive

∆SX = SXf − SXi≤ SXf + S[pXf ||pXeq,f ]− SXi= β(∆〈HX 〉 −∆FX ). (3.89)


β(∆〈HX 〉+ ∆〈HY〉 −∆FX −∆FY +Q) ≥ IX :M − SM. (3.90)

Since the energy change of the total system ∆〈HX 〉+ ∆〈HY〉+Q should be identified as

53

work W , we conclude

W −∆FX −∆FY ≥ kBT (IX :M − SM). (3.91)

This equality presents the minimum work needed to perform the measurement, which

may be interpreted as a rigorous formulation of Brillouin’s argument.

Erasure process

Next, we consider an erasure process of the stored information. We assume that the total

system of this process consists of the memory Y and the heat bath B, and the total system

is subject to the Hamiltonian dynamics. We assume that the Hamiltonian is written as

HY×B(y, b, λ) = HY(y, λ) +HB(b) +H int(y, b, λ), (3.92)

where H int is the interaction term, which is assumed to vanish at the initial and final

times.

Since the total system is under Hamiltonian dynamics, we obtain

SY×Bi = SY×Bf . (3.93)

We assume that the initial state is a product state as

pY×Bi (yi, bi) = pYi (yi)pBeq(bi), (3.94)

and we obtain

SY×Bi = SYi + β(〈HB〉i − FB). (3.95)

By the same procedure to derive Eq. (3.48), we obtain

SY×Bf ≤ SYf + β(〈HB〉f − FB). (3.96)


∆SY + βQ ≥ 0, (3.97)

where the dissipated heat is defined as

Q = 〈HB〉f − 〈HB〉i. (3.98)

54

We assume that the memory initially stores a classical probability distribution p(m)

and it is in the local equilibrium state of Ym under the condition that the stored infor-

mation is m, i.e.,

pY(yi) =∑m∈M

p(m)pYmeq (yi, λi). (3.99)

Therefore, the initial entropy of the memory is calculated as

SYi =

∫Ydyi

∑m∈M

p(m)pYmeq (yi, λi) ln∑m′∈M

p(m′)pYm′eq (yi, λi)

=∑m∈M

∫Ymdyi p(m)pYmeq (yi, λi) ln

∑m′∈M

p(m′)pYm′eq (yi, λi)

=∑m∈M

∫Ymdyi p(m)pYmeq (yi, λi) ln p(m)pYmeq (yi, λi)

= SM + β(〈HY〉i − FYi ). (3.100)

On the other hand, by the positivity of the relative entropy, we obtain

SYf ≤ β(〈HY〉f − FYf ), (3.101)

which reduces Eq. (3.97) to

β(∆〈HY〉 −∆FY +Q) ≥ SM. (3.102)

We identify ∆〈HY〉 + Q as work W , since it is the energy increase of the total system.

Therefore, we conclude

W −∆FY ≥ kBTSM, (3.103)

which reveals the minimum work needed to erase the information stored in the memory.

When FYm(λf) = FY0(λi) for arbitrary m, we have no free-energy difference and


W ≥ kBTSM. (3.104)

In other words, in symmetric memories, Eq. (3.102) reduces to Landauer’s principle, which

states that the minimum cost for erasure is the Shannon entropy of the stored information.

55

3.2.4 Reconciliation of the demon with the conventional second

law

In this section, we summarize the results obtained in the previous sections, and demon-

strate that the second law is recovered in the total process [19].

The second law under feedback control is

Wfb −∆FXfb ≥ −kBTIX :M. (3.105)

The minus of the right-hand side quantifies the energy gain of feedback control, or the

extractable work beyond the conventional second law thanks to Maxwell’s demon. In the

measurement process, we have obtained

Wmeas −∆FXmeas −∆FYmeas ≥ kBT (IX :M − SM), (3.106)

whose right-hand side represents the additional work cost needed for the demon to perform

the measurement. In the erasure process, we have derived

Weras −∆FYeras ≥ kBTSM. (3.107)

The right-hand side is the additional work cost for the erasure of the information stored

by the demon. Therefore, the sum of the costs for both the measurement and erasure is

given by

Wm+e −∆FXm+e −∆FYm+e ≥ kBTIX :M, (3.108)

where we define Wm+e = Wmeas +Weras and other quantities in a similar way. We see that

the work gain by the demon is precisely compensated by the work cost that the demon

pays for the measurement and erasure. In fact, in total, the information quantities on the

right-hand sides of Eqs. (3.105) and (3.108) are cancelled out, and we obtain

Wtot −∆FXtot −∆FYtot ≥ 0, (3.109)

where we define Wtot = Wmeas + Wfb + Weras and other quantities in a similar way. The

conventional second-law-like inequality (3.109) is recovered in the total process consisting

of the measurement, feedback control, and erasure processes. In particular, in an isother-

mal cycle, Eq. (3.109) reduces to Wtot ≥ 0, which is nothing but Kelvin’s principle for the

isothermal composite system consisting of the system and the memory.

As reviewed in Sec. 2.1, Brillouin argued that the entropy reduction by the demon is

56

compensated by the cost for the measurement. On the other hand, Landauer’s principle

says that the work cost for the erasure of the stored information exceeds the work gain

of the feedback control. Although these views are true for some specific systems, they

are not generally true. What compensates the work gain by the demon in general is

not the individual work cost for the measurement or erasure but the joint work cost for

the measurement and erasure processes. The information-thermodynamic inequalities

summarized above reveal that the reconciliation of Maxwell’s demon with the second law

is achieved in a rigorous manner.

In terms of the Shannon entropy, we have obtained

∆SXfb + βQfb ≥ −IX :M, (3.110)

∆SXmeas + ∆SYmeas + βQmeas ≥ IX :M, (3.111)

∆SYeras + βQeras ≥ 0, (3.112)

∆SXm+e + ∆SYm+e + βQm+e ≥ 0, (3.113)

∆SXtot + ∆SYtot + βQtot ≥ 0. (3.114)

The inequalities (3.111) and (3.112) are simpler than Eqs. (3.106) and (3.107) in that

they do not have the Shannon entropy term SM explicitly. We note that the inequalities

on the Shannon entropy (3.110), (3.111), (3.112), (3.113), and (3.114) are stronger than

the inequalities on the work (3.105), (3.106), (3.107), (3.108), and (3.109), since the latter

inequalities can be derived from the former inequalities based on positivity of the relative

entropy.

3.3 Nonequilibrium equalities under measurements

and feedback control

The second-law-like inequality under measurements and feedback control derived in the

previous section can be generalized to nonequilibrium equalities as the conventional second

law is generalized to the nonequilibrium equalities reviewed in Sec 2.1.

In this section, we review information-thermodynamic nonequilibrium equalities. Then,

we derive these equalities in a unified manner similar to that of Sec. 2.2.

3.3.1 Information-thermodynamic nonequilibrium equalities

In 2010, Sagawa and Ueda generalized the Jarzynski equality in a Markov stochastic

system under feedback control based on a single measurement and obtained the Sagawa-

57

Ueda equality [20]

〈e−β(W−∆F )−I〉 = 1, (3.115)

where I is the unaveraged mutual information obtained by the measurement. Using

Jensen’s inequality, we succinctly reproduce the second law of information thermodynam-

ics as

〈W 〉 −∆F ≥ −kBT 〈I〉. (3.116)

Later, the Sagawa-Ueda equality is generalized to Markov systems with multiple mea-

surements [24], and to non-Markov systems with multiple measurements [59].

The Sagawa-Ueda equality has variants as the Jarzynski equality has the variants

reviewed in Sec. 2.1. The Hatano-Sasa relation is generalized to systems under feedback

control as [60]

〈e−∆φ−∆sex/kB−I〉 = 1. (3.117)

The associate inequality is given by

〈∆φ〉+ 〈∆sex〉/kB ≥ −〈I〉. (3.118)

Moreover, in Ref. [60], the Seifert relation is generalized to

〈e−∆stot/kB−I〉 = 1, (3.119)

which leads to a second-law-like inequality

〈∆stot〉 ≥ −kB〈I〉. (3.120)

Reference [61] derived

〈e−∆shk−I〉 = 1, (3.121)

and

〈∆shk〉 ≥ −kB〈I〉. (3.122)

Equations (3.115), (3.117), (3.119), and (3.121) are summarized in terms of the formal

58

entropy production σ as

〈e−σ−I〉 = 1. (3.123)

This equality is a general nonequilibrium equalities under measurements and feedback

control. The second law of information thermodynamics is reproduced as

〈σ〉 ≥ −〈I〉. (3.124)

3.3.2 Derivation of information-thermodynamic nonequilibrium

equalities

In this section, we derive the nonequilibrium equalities under measurements and feedback

control.

Setup

We consider a nonequilibrium process in a classical non-Markov stochastic system with

feedback control from time ti to tf . To formulate the dynamics of the system, we discretize

the time interval into N parts and define tn = ti +n(tf− ti)/N (n = 0, · · · , N). Let X and

Y denote phase spaces of the system and the memory, respectively. Moreover, we denote

the phase-space positions in X and Y at time tn by xn and yn, respectively. An external

parameter to control the system is denoted by λ, and the value of λ at time tn is λn.

Initially, the system and memory are at (x0, y0) sampled from an initial joint probability

distribution p0(x0, y0). Then, the system evolves from time t0 to t1 and a measurement

is performed at time t1 to obtain a measurement outcome y1. After that, the system is

subject to feedback control driven by an external parameter λ1(y1) from time t1 to t2. In

this way, we repeat measurements and feedback control. We define Xn = (x0, x1, · · · , xn)

and Yn = (y0, · · · , yn), and therefore XN and YN−1 represents the entire trajectory of

the system and memory, respectively. Since λn is adjusted based on the measurement

outcomes before time tn, λn is a function of Yn. Therefore, the non-Markov dynamics of

the system is determined by transition probabilities

p(xn+1|Xn, λn(Yn)). (3.125)

On the other hand, since the measurement outcome depends on the trajectory of the

system before the measurement, the dynamics of the memory is determined by

p(yn|Xn). (3.126)

59

Therefore, the joint probability to realize XN and YN−1 is calculated as

P [XN , YN−1] =N−1∏n=1

p(xn+1|Xn, λn(Yn))p(yn|Xn) · p(x1|x0, λ0(y0))p0(x0, y0)

= P tr[XN |x0,ΛN−1(YN−1)]P tr[YN−1|XN−1]p0(x0, y0), (3.127)

where we define the transition probabilities as

P tr[XN |x0,ΛN−1(YN−1)] =N−1∏n=0

p(xn+1|Xn, λn(Yn)), (3.128)

P tr[YN−1|XN−1] =N−1∏n=1

p(yn|Xn). (3.129)

Dividing Eq. (3.127) by P [YN−1], we calculate the conditional probability as

P [XN |YN−1] = P tr[XN |ΛN−1(YN−1)]P tr[YN−1|XN−1]p0(x0, y0)

P [YN−1]p0(x0). (3.130)

Therefore, the conditional probability is different from the transition probability under

measurements and feedback control.

Following Ref. [59], we define the information obtained by the measurement at time tn

as the mutual information between yn and Xn under the condition that we have obtained

Yn−1, that is,

In[yn, Xn|Yn−1] = − ln p(yn|Yn−1) + ln p(yn|Xn, Yn−1)

= − ln p(yn|Yn−1) + ln p(yn|Xn). (3.131)

The sum of the mutual information is calculated as

I =n−1∑n=1

In = − lnP [YN−1|y0] + lnP tr[YN−1|XN−1]

= − lnP [YN−1]

P tr[YN−1|XN−1]p0(y0), (3.132)

which is the total information obtained by all the measurements. Using the total mutual

information, we can transform Eq. (3.130) as

P tr[XN |ΛN−1(YN−1)]

P [XN |YN−1]= e−I−Ii , (3.133)

60

where Ii is the mutual information at the initial time defined by

Ii = − lnp0(x0)p0(y0)

p0(x0, y0). (3.134)

Derivation of information-thermodynamic nonequilibrium equalities

The formal entropy production σ is defined as the ratio of the reference probability to the

original probability under a fixed protocol as in Eq. (2.32). Therefore, we obtain

P r[XN |ΛN−1(YN−1)]

P tr[XN |ΛN−1(YN−1)]= e−σ[XN |ΛN−1(YN−1)]. (3.135)

We define the joint probability of the reference process as

P r[XN , YN−1] = P r[XN |ΛN−1(YN−1)]P [YN−1], (3.136)

which means that we sample the reference process conditioned by YN−1 with the same

probability P [YN−1] as the original process. From Eqs. (3.127) and (3.136), the ratio of

the reference joint probability to the original joint probability is calculated as

P r[XN , YN−1]

P [XN , YN−1]=

P r[XN |ΛN−1(YN−1)]

P tr[XN |ΛN−1(YN−1)]

P [YN−1]p0(x0)

P tr[YN−1|XN−1]p0(x0, y0)

= e−σ−I−Ii . (3.137)

Multiplying both sides by P [XN , YN−1] and integrating over XN and YN−1, we obtain

〈e−σ−I−Ii〉 = 1, (3.138)

because the reference probability is normalized to unity. In ordinary feedback protocols,

the system is assumed not to be correlated with the memory at the initial time. Therefore,


〈e−σ−I〉 = 1. (3.139)

The appropriate choices of the reference probability explained in Sec. 2.2 reduce Eq. (3.139)

to Eqs. (3.115), (3.117), (3.119), and (3.121).

3.4 Experiments

In this section, we briefly review experimental demonstrations of Maxwell’s demon.

61

(a) (b)

Figure 3.6: (a) Two washboard potentials induced by an electromagnetic field. Ifthe Brownian particle climbs up the potential due to thermal agitation, the potential isswitched to the other potential to prevent the particle from descending. (b) Free-energygain of the Brownian particle subtracted by the amount of work done on it. The abscissarepresents the delay time ε it takes to switch the potential after the position measurement.The first two data points show the net free energy gain beyond the conventional secondlaw of thermodynamics. Reproduced from Figs. 2b and 3d in Ref. [16]. Copyright 2010by the Macmillan Publishers Limited.

The first experimental realization of Maxwell’s demon was done by Toyabe et al. [16].

They demonstrated that the free energy of a Brownian particle can be increased by feed-

back control based on measurements of the position of the particle. A couple of polystyrene

beads are suspended in a water with one of them anchored to a glass plate. The other

bead can move on a ring, and is subjected to a washboard potential created by four

electrodes (see Fig. 3.6 (a)). At a certain instant in time, the position of the particle is

measured. After a delay time ε, if the particle climbs up the potential due to thermal

agitation, the potential is switched to the other potential to prevent the particle from

descending; otherwise the potential is left unchanged. This protocol of feedback control

is repeated. As a result, the particle is able to gain free energy larger than the work done

on it in this feedback-controlled process (see Fig. 3.6 (b)), namely, it was demonstrated

that information obtained by the measurements can be used as a resource for free energy.

The information-thermodynamic nonequilibrium equality (3.115) was experimentally

verified in Ref. [17]. A feedback-controlled two-state system similar to the Szilard engine

was implemented in a single-electron box (SEB) illustrated in Fig. 3.7 (a). The gate

voltage Vg in Fig. 3.7 (a) is adjusted so that the minimum charging energy is achieved

when the average number n of the electrons that have tunneled from the left island to

62

(a) (b)

Figure 3.7: (a) Schematic illustration of the system demonstrating the information-thermodynamic nonequilibrium equality (3.115). The single-electron box (SEB) consistsof two metallic islands connected by a junction, through which electrons are transportedby tunneling. The number n of excess electrons in the right island is monitored by a single-electron transistor (SET). The ground-state average value ng of n is controlled by the gatevoltage Vg. (b) Experimental verification of Eq. (3.115). The abscissa represents the errorrate of the measurement of n. The data shown by black squares deviate significantly fromunity, showing the breakdown of the Jarzynski equality, whereas the red squares showthat Eq. (3.115) holds. Reproduced from Figs. 1 (a) and 3 (b) in Ref. [17]. Copyright2014 by the American Physical Society.

the right one is ng = 0.5. The temperature of the system is low enough that the SEB is

in either the n = 0 or n = 1 state. Therefore, the state with n = 0 is realized with the

probability of 0.5, so is the state with n = 1. The state of the SEB is monitored by a

single-electron transistor (SET) (see Fig. 3.7 (a)), and the feedback control is conducted

by changing Vg based on the value of n to extract work from the SEB. The average in

Eq. (3.115) was experimentally confirmed to be unity within experimental errors as shown

in Fig. 3.7 (b).

63

Chapter 4

Nonequilibrium Equalities in

Absolutely Irreversible Processes

As discussed in Chap. 2, nonequilibrium equalities apply to rather general nonequilibrium

situations. However, it is known that integral fluctuation theorems are inapplicable to

some situations. We propose a new concept of absolute irreversibility as a novel class of

irreversibility that encompasses the entire range of those situations to which conventional

integral nonequilibrium equalities cannot apply. In mathematical terms, the absolute

irreversibility is defined as the singular part of the reference probability measure, and

is uniquely separated from the ordinary irreversible part by Lebesgue’s decomposition

theorem [25, 26]. We derive nonequilibrium equalities that are applicable to absolutely

irreversible processes based on measure theory. Inequalities derived from our nonequi-

librium equalities give a positive-definite lower bound of the entropy production when a

process involves absolute irreversibility.

First of all, we consider free expansion to illustrate that absolute irreversibility causes

inapplicability of conventional nonequilibrium integral equalities, and define absolute ir-

reversibility in terms of measure theory. Then, we derive nonequilibrium equalities in

absolutely irreversible processes based on Lebesgue’s decomposition theorem. Next, we

verify our nonequilibrium equalities in several examples. Finally, we compare our method

with a conventional method and discuss merits of ours.

In this chapter, we restrict our attention to systems without measurements and feed-

back control. This chapter is mainly based on Ref. [27].

64

Figure 4.1: (a) Forward process of free expansion. An ideal single-particle gas is ini-tially in a local equilibrium in the left box with temperature T . Then, the partition isremoved, and the gas freely expand to the entire box. (b) Backward process of free ex-pansion. Initially, the single-particle gas is in a global equilibrium of the entire box withtemperature T . Then, the partition is inserted, and the gas particle is in either the leftor right box. The backward path ending in the right box (indicated by the blue arrow)has no corresponding forward path. Therefore, this is a singular path with a negativelydivergent entropy production. Reproduced from Fig. 1 of Ref. [27]. Copyright 2014 bythe American Physical Society.

4.1 Inapplicability of conventional integral nonequi-

librium equalities and absolute irreversibility

In this section, we introduce absolute irreversibility in an example of free expansion, to

which the Jarzynski equality cannot apply. Then, we mathematically define absolute

irreversibility in terms of measure theory.

4.1.1 Inapplicability of the Jarzynski equality

The Jarzynski equality is known to be inapplicable to free expansion [21, 22]. Here, we

illustrate this fact. Suppose that an ideal single-particle gas at temperature T is prepared

in the left side of a box with a partition as illustrated in Fig. 4.1 (a). Then, we remove the

partition and let the gas expand to the entire box. In this process, work is not extracted

(W = 0), whereas the free energy decreases (∆F < 0). Therefore, the dissipated work is

always positive:

W −∆F > 0. (4.1)

65

Thus, we have

〈e−β(W−∆F )〉 < 1, (4.2)

which means that the Jarzynski equality (2.4) is not satisfied in this process [21]. In

physical terms, this is because the Jarzynski equality assumes that the initial state is

in a global equilibrium and this assumption is not satisfied in the present case. Recall

that, in free expansion, the initial state is not a global equilibrium state, but only a local

equilibrium state [22]. Therefore, the Jarzynski equality cannot apply to free expansion.

Then, it is natural to ask why a local equilibrium state cannot be assumed as an initial

condition of the conventional integral nonequilibrium equality.

The mathematical reason is that we have paths with a divergent entropy production

when we start from a local equilibrium state. This statement is illustrated in free expan-

sion as follows. We consider a set of virtual paths ΓR starting from the right box. By

assumption, the probability of these paths in the forward process vanishes: P [ΓR] = 0.

On the other hand, the probability of the corresponding backward paths is nonvanishing:

P†[Γ†R] 6= 0. Therefore, we have

∃Γ, P [Γ] = 0 & P†[Γ†] 6= 0. (4.3)

For these paths, the entropy production is negatively divergent in the context of the

Crooks fluctuation theorem (2.12) because

σ = β(W −∆F ) = − lnP†[Γ†]P [Γ]

→ −∞ (4.4)

and

e−σ =P†[Γ†]P [Γ]

→∞. (4.5)

Due to the paths with a divergent entropy production, the conventional integral nonequi-

librium equality breaks down:

〈e−σ〉 =

∫P[Γ] 6=0

e−σP [Γ]DΓ

=

∫P[Γ] 6=0

P†[Γ†]DΓ†

= 1−∫P[Γ]=0

P†[Γ†]DΓ†

< 1. (4.6)

66

Therefore, we conclude that the negatively divergent entropy production of the paths

starting from the region in which the initial probability vanishes is what makes the con-

ventional integral nonequilibrium equality inapplicable to the process starting from a local

equilibrium state.

This situation [Eq. (4.3)] makes a stark contrast to ordinary irreversible processes,

where every backward path has the corresponding forward path with a nonvanishing

probability:

∀Γ, P†[Γ†] 6= 0⇒ P [Γ] 6= 0, (4.7)

or

∀Γ, P [Γ] = 0⇒ P†[Γ†] = 0. (4.8)

Therefore, in the ordinary irreversible case, the exponentiated entropy production remains

finite

e−σ =P†[Γ†]P [Γ]

<∞, (4.9)

and the thermodynamic irreversibility is quantitatively characterized by the entropy pro-

duction. Although these paths are thermodynamically irreversible, they are stochastically

reversible in a sense of Eq. (4.7), namely, every backward path has the nonvanishing for-

ward counterpart. In contrast, if the condition (4.3) holds, there exist backward paths

that have no counterparts in the forward process, which means that these paths are not

even stochastically reversible. Therefore, we shall call these paths absolutely irreversible

paths, and call the processes with absolutely irreversible paths absolutely irreversible

processes.

Mathematically, probability theory is based on measure theory, and the ratio P†/P is

interpreted as the transformation function of the two probability measures. Therefore, we

need measure theory to formulate absolute irreversibility. We thus give a mathematical

definition of absolute irreversibility.

4.1.2 Definition of absolute irreversibility

Let M[DΓ] denote the probability measure of the original process. This is a generaliza-

tion of the description in terms of the probability density which is written as P [Γ]DΓ.

Let the probability measure of the reference process be denoted by Mr[DΓ] which is a

generalization of Pr[Γ]DΓ. According to Lebesgue’s decomposition theorem [25, 26], the

67

/71京大基研セミナー 2014年3月5日 /15物理学会春季大会 2014年3月29日

Lebesgueの分解定理

1

Phase Space

Prob. Density

Figure 4.2: Schematic illustration of Lebesgue’s decomposition theorem. The abscissarepresents coordinates of phase space and the ordinate shows the probability density. Ver-tical lines represent δ-function-like localization. The reference probability measure Mr

(blue solid curve) is decomposed into two parts with respect to the original probabilitymeasureM (dashed curve). The probability ratio is well-defined in the absolutely contin-uous part, whereas it diverges in the singular part. Note that the δ-function-like referenceprobability measure is absolutely continuous if its singular part coincides with that of theoriginal probability measure as shown in the middle figure. Reproduced from Fig. 2 ofRef. [27]. Copyright 2014 by the American Physical Society.

reference probability measure can be uniquely decomposed into two parts as

Mr =MrAC +Mr

S, (4.10)

where MrAC and Mr

S are absolutely continuous and singular with respect to M, respec-

tively (see Fig. 4.2). The absolute continuity of MrAC guarantees that the probability

ratio is well-defined due to the Radon-Nikodym theorem [25, 26] as

DMrAC

DM , (4.11)

which is an integrable function with respect to M. In physical terms, it is this ratio

that gives the entropy production. Therefore, in measure theory, the Crooks fluctuation

theorem reads

DMrAC

DM = e−σ. (4.12)

On the other hand, the probability defined by MrS takes a nonzero value in the region

where the probability defined by M vanishes. Therefore, the ratio of MrS to M is di-

vergent, and we cannot define a finite entropy production through this ratio. Thus, in

physical terms, MrS corresponds to the absolutely irreversible part. We therefore iden-

tify MrAC as the ordinary irreversible part and Mr

S as the absolutely irreversible part.

If MrS does not vanish, the conventional integral nonequilibrium equality breaks down

[21, 22, 62, 63]. See Appendix B for the mathematical definitions of absolute continuity

and singularity, and a mathematical statement of Lebesgue’s decomposition theorem.

68


Lebesgueの分解定理

1

Phase Space

Prob. Density

Figure 4.3: Schematic illustration of the stronger version of Lebesgue’s decompositiontheorem. The reference probability measure Mr (blue solid curve) is decomposed intothree parts with respect to the original probability measure M (dashed curve). In theabsolutely continuous part (bottom left), the probability ratio of Mr

ac to M is well-defined. In the singular continuous part (bottom middle), the probability ratio ofMr

sc toM is divergent because the probability density of M vanishes and that of Mr

sc remainsnonvanishing. In the discrete part (bottom right), the probability ratio of Mr

d to M isdivergent because the probability density of Mr

d is divergent whereas M remains finite.We note thatM does not involve δ-function-like localization, which is the assumption ofthe stronger version of Lebesgue’s decomposition and needed for the uniqueness of thisdecomposition (see Appendix B for more detail). Reproduced from Fig. 3 of Ref. [27].Copyright 2014 by the American Physical Society.

When M can be written in terms of a probability density, a stronger version of

Lebesgue’s decomposition theorem holds. In this case, Mr is decomposed into three

parts as

Mr =Mrac +Mr

sc +Mrd, (4.13)

where Mrac and Mr

sc are absolutely continuous and singular continusous with respect

to M, respectively, and Mrd is the discrete part of Mr (see Fig. 4.3). The singular

continuous part Mrsc corresponds to the region where the probability ratio is divergent

because the denominator, which is the forward probability, vanishes. This term represents

the effect of free expansion. The discrete partMrd has δ-function-like localization, and the

probability ratio is divergent because the numerator, which is the reference probability,

diverges. This part arises when particles can localize and do not undergo thermal diffusion;

such a situation occurs when there are trapping centers of particles. In this way, the

absolute irreversibility is classified into two categories. The correspondence between the

classification of irreversibility and that of probability measure is summarized in Table 4.1.

69

4.2 Nonequilibrium equalities in absolutely irreversible

processes

In this section, we derive nonequilibrium equalities applicable to absolutely irreversible

processes based on Lebesgue’s decomposition theorem (4.10) and (4.13).

First, we derive general nonequilibrium equalities in absolutely irreversible processes.

Then, we show that they reduce to several individual nonequilibrium equalities with their

specific meanings of the entropy production by proper choices of the reference probability

distribution as described in Sec. 2.2.

4.2.1 General formulation

First, we derive nonequilibrium equalities based on Lebesgue’s decomposition theorem (4.10).

Let F [Γ] denote an arbitrary functional of a path Γ, and let 〈· · ·〉, 〈· · ·〉r and 〈· · ·〉rI (I=AC,

S) denote the average overM,Mr andMrI, respectively. From Lebesgue’s decomposition

theorem (4.10), we obtain

〈F〉r = 〈F〉rAC + 〈F〉rS. (4.14)

On the other hand, using the Radon-Nikodym derivative (B.8), we evaluate the average

over the absolutely continuous part in terms of the entropy production as

〈F〉rAC =

∫F [Γ]Mr

AC[DΓ]

=

∫F [Γ]

DMrAC

DM

∣∣∣∣Γ

M[DΓ]

=

∫F [Γ]e−σM[DΓ]

= 〈Fe−σ〉, (4.15)

Table 4.1: Correspondence between the classification of irreversibility and that of prob-ability measure.

Class of irreversibility Ordinary Absolute I Absolute II

Class of measure absolutely continuous singular continuous discrete

Exponentiated entropy: e−σPr[Γ]

P [Γ]=finite

Pr[Γ]

0=∞ δ(0)

P [Γ]=∞

70

where we use the Crooks fluctuation theorem (4.12) to obtain the third line. Therefore,

by Eqs. (4.14) and (4.15), we obtain

〈Fe−σ〉 = 〈F〉r − 〈F〉rS, (4.16)

which can be regarded a generalization of the master fluctuation theorem (2.35). If we

set F to unity, we obtain

〈e−σ〉 = 1− λS, (4.17)

where

λS =

∫Mr

S[DΓ] (4.18)

is the probability of the singular part. We note that λS is uniquely defined because of

the uniqueness of Lebesgue’s decomposition (4.10). As explained in the previous section

and summarized in Table 4.1, the absolute irreversibility has two classes. Therefore, λS

is calculated as the sum of two contributions of these classes of absolute irreversibility.

One is the probability of those reference paths whose corresponding original paths have

vanishing probability. The other contribution is the probability of localized reference

paths. To the best of our knowledge, the localized contribution had not been considered

before our research [27]. However, this part renders the conventional nonequilibrium

equalities inapplicable. Namely, only when λS = 0, we reproduce 〈e−σ〉 = 1.

Using Jensen’s inequality 〈e−σ〉 ≥ e−〈σ〉, we obtain

〈σ〉 ≥ − ln(1− λS). (4.19)

Therefore, the second-law-like inequality 〈σ〉 ≥ 0 is valid even with absolute irreversibil-

ity, because the right-hand side of Eq. (4.19) is nonnegative. Moreover, if there exists

absolute irreversibility λS > 0, then Eq. (4.19) imposes a stronger restriction on the en-

tropy production than the conventional inequality because the right-hand side will then

be strictly positive. Thus, the average of the entropy production must be strictly positive

in absolutely irreversible processes.

When the stronger version of Lebesgue’s decomposition holds, by following the same

procedure as before, we obtain

〈Fe−σ〉 = 〈F〉r − 〈F〉rsc − 〈F〉rd, (4.20)

〈e−σ〉 = 1− λsc − λd, (4.21)

71

where 〈· · · 〉ri (i=sc, d) represents the average overMri , and λsc and λd are the probabilities

of the singular continuous part and the discrete part, respectively. We can calculate λsc

as the sum of the probabilities of those reference paths whose corresponding counterparts

in the original process vanish. On the other hand, λd is calculated as the sum of localized

reference paths. From Eq. (4.21), we obtain an inequality

〈σ〉 ≥ − ln(1− λsc − λd), (4.22)

which leads to a fundamental lower bound on the entropy production in absolutely irre-

versible processes.

4.2.2 Physical implications

We apply the general results in the previous section to processes starting from a restricted

region.

Here, we consider time reversal as the reference dynamics. In accordance with Ta-

ble 2.1, in a Langevin or Hamiltonian system, Eq. (4.12) reduces to

e−σ = e−βQdµL,AC

dµ0

∣∣∣∣Γ0

dµ†0,AC

dµL

∣∣∣∣∣Γτ

, (4.23)

where we rewrite the boundary term in terms of measure theory and Γ represents the

configuration coordinates in the case of an overdamped Langevin system and the phase

space coordinates of the system and heat bath in the case of a Hamiltonian system. Here,

µ0, µ†0 and µL represent the initial probability measure of the original process, the initial

probability measure of the time-reversed process and the Lebesgue measure on the phase

space Ω, respectively, and µL,AC is the absolutely continuous part of µL with respect to

µ0 and µ†0,AC is the absolutely continuous part of µ†0 with respect to µL.

Generalization of the Jarzynski equality

Now, let us assume that the initial state of the original process is a local equilibrium state

which is restricted to a region D0 ⊂ Ω as

µ0(dΓ0) = e−β(H(Γ0,λ0)−F0)χD0(Γ0)µL(dΓ0), (4.24)

where χD0(Γ0) is the characteristic function defined by

χD0(Γ0) =

1 Γ0 ∈ D0

0 Γ0 /∈ D0,(4.25)

72

and the free energy is defined by

e−βF0 =

∫D0

e−βH(Γ0,λ0)µL(dΓ0). (4.26)

In this case, the absolutely continuous part of µL with respect to µ0 is

µL,AC(dΓ0) = χD0(Γ0)µL(dΓ0). (4.27)


dµL,AC

dµ0

∣∣∣∣Γ0

= eβ(H(Γ0,λ0)−F0). (4.28)

On the other hand, we set the initial probability measure of the time-reversed process to

a local equilibrium distribution in a region Dτ as

µ†0(dΓτ ) = e−β(H(xτ ,λτ )−Fτ )χDτ (Γτ )µL(dΓτ ), (4.29)

which is already absolutely continuous with respect to µL, namely, µ†0,AC = µ†0. Therefore,

we obtain

dµ†0,AC

dµL

∣∣∣∣∣Γτ

= e−β(H(xτ ,λτ )−Fτ )χDτ (Γτ ). (4.30)

Then, substituting Eqs. (4.28) and (4.30) into Eq. (4.23), we obtain

e−σ = e−β(Q+∆H−∆F )χDτ (Γτ )

= e−β(W−∆F )χDτ (Γτ ), (4.31)

where we use the first law of thermodynamics to obtain the last equality. Thus, Eq. (4.17)

reduces to

〈e−β(W−∆F )χDτ (Γτ )〉 = 1− λS. (4.32)

In particular, if we set Dτ to Ω, we obtain

〈e−β(W−∆F )〉 = 1− λS, (4.33)

which is a generalization of the Jarzynski equality (2.4).

73

Generalization of the Seifert relation

Here, we assume that the initial probability distribution of the original process is abso-

lutely continuous with respect to the Lebesgue measure, and therefore we have

µ0(dΓ0) = p0(Γ0)µL(dΓ0). (4.34)

Let D0 denote the support of p0. Then, the absolutely continuous part of µL with respect

to µ0 is

µL,AC(dΓ0) = χD0(Γ0)µL(dΓ0). (4.35)

Therefore, the Radon-Nikodym derivative is

dµL,AC

dµ0

∣∣∣∣Γ0

=χD0(Γ0)

p0(Γ0). (4.36)

We set the initial probability distribution of the reference process to be equal to the final

probability distribution of the original process: µ†0 = µτ . Then, the Radon-Nikodym

derivative

dµ†0,AC

dµL

∣∣∣∣∣Γτ

=dµτ,AC

dµL

∣∣∣∣Γτ

= pτ (Γτ ) (4.37)

represents the (unnormalized) final probability distribution of the original process. There-

fore, Eq. (4.23) reduces to

e−σ = e−βQpτ (Γτ )

p0(Γ0)χD0(Γ0)

= e−∆sbath/kBe−∆s/kBχD0(Γ0). (4.38)

Thus, Eq. (4.17) reduces to

〈e−∆stot〉 = 1− λS, (4.39)

where we use χD0(Γ0)p0(Γ0) = p0(Γ0) because D0 is the support of p0. This is a general-

ization of the Seifert relation (2.25).

Up to here, we set the reference dynamics to the time-reversed dynamics, and have shown

that our nonequilibrium equality (4.17) with absolute irreversibility derives generalized

integral nonequilibrium equalities (4.33) and (4.39) involving the entropy production with

74

Table 4.2: Correspondence between reference probabilities, specific meanings of theentropy production and nonequilibrium equalities with absolute irreversibility.

Reference dynamics Ref. ini. state Entropy production Nonequilibrium equality

time reversal canonical dissipated work 〈e−β(W−∆F )〉 = 1− λS

time reversal final state total 〈e−∆stot/kB〉 = 1− λS

time reversal initial state dissipation functional 〈e−Ω〉 = 1− λS

dual initial state housekeeping 〈e−∆shk/kB〉 = 1− λS

time-reversed dual steady state excess 〈e−∆φ−∆sex/kB〉 = 1− λS

the heat bath. We note that our nonequilibrium equality (4.17) also derives generalized

integral fluctuation theorems for the housekeeping and excess entropy production in a

similar manner under the proper choices of the reference dynamics as summarized in

Table 4.2.

4.3 Examples of absolutely irreversible processes

In this section, we verify our nonequilibrium equality in three examples: free expansion,

an overdamped Langevin process starting from a local equilibrium, and an overdamped

Langevin system with a trapping center.

4.3.1 Free expansion

First of all, we discuss the case of free expansion (see Fig. 4.1 (a)). Initially, an ideal single-

particle gas is confined in the left box with temperature T . The entire box is assumed

to be divided in the volume ratio l : 1 − l by a partition. Since the initial state can be

described by the probability density, the stronger version of Lebesgue’s decomposition

holds. Thus, Eq. (4.33) reduces to

〈e−β(W−∆F )〉 = 1− λsc − λd. (4.40)

We remove the partition, and the gas expands to the entire box. In this process, work is

not extracted: W = 0, whereas the free energy of the gas decreases due to the expansion

by ∆F = kBT ln l(< 0). Therefore, the left-hand side of Eq. (4.40) is calculated to be

〈e−β(W−∆F )〉 = l. (4.41)

We consider the time-reversed process of this free expansion to calculate the absolute

75

irreversible probabilities (see Fig. 4.1 (b)). In the time-reversed process, the system is

in a global equilibrium at the initial time. Then, the partition is inserted at the same

position as the original process. The gas particle is in either the left or the right box.

The paths ending in the right box have no corresponding forward paths in the original

process. Therefore, these paths are singular continuous paths. The probability of these

paths is proportional to the volume fraction of the right box, and therefore the singular

continuous probability is

λsc = 1− l. (4.42)

On the other hand, since there is no discrete path, which is a single path with a finite

positive probability, we have

λd = 0. (4.43)


1− λsc − λd = l. (4.44)

Thus, our nonequilibrium equality (4.40) is verified for the case of free expansion of an

ideal single-gas particle.

4.3.2 Process starting from a local equilibrium

Next, we consider an overdamped Langevin process starting from a local equilibrium

state. The Langevin particle is confined in a one-dimensional ring. The potential consists

of n identical harmonic potential wells with the same stiffness (spring constant) k(t) as

illustrated in Fig. 4.4 (a). Initially, the system is prepared in a local equilibrium state with

temperature T in a given well, and therefore the initial probability vanishes elsewhere.

We subject the system to a nonequilibrium process during a time interval τ . We decrease

the stiffness of the potentials from k = K to 0 at a constant rate between t = 0 and τ/2,

and then increase it from k = 0 to n2K at a constant rate between t = τ/2 and τ . Since

the initial state can be written in terms of the probability density, the stronger version of

Lebesgue’s decomposition holds. Therfore, Eq. (4.33) reduces to

〈e−β(W−∆F )〉 = 1− λsc − λd. (4.45)

Now, we consider the time-reversed process to calculate the singular probabilities.

Initially, the system is in a global equilibrium at temperature T with stiffness k = n2K.

76

(a) (b)

(c) (d)

Figure 4.4: (a) Schematic illustration of an overdamped system consisting of n identicalharmonic potentials confined in a one-dimensional ring. In the original process, the systemis in a local equilibrium in a given well at the initial time. (b) Probability density ofwork performed on the system for several n values. The triangles indicate the points ofW = kBT lnn. (c) Values of 〈e−βW 〉 at each n. Superimposed is the 1/n curve (withno free parameters). (d) Values of 〈βW 〉 at each n. The line represents the minimumdissipation given by Eq. (4.50). The parameters are chosen as follows: diffusion constantD = 10−13m2/s; temperature T = 300K; duration of the process τ = 10sec; half widthof a single potential a = 10−6m; the initial stiffness of the potential K is chosen tosatisfy Ka2/2 = 5kBT . The statistical average is obtained from 106 samples for each n.Reproduced from Fig. 4 of Ref. [27]. Copyright 2014 by the American Physical Society.

77

The stiffness is decreased from k = n2K to 0 between t = 0 and τ/2, and then increased

from k = 0 to K between t = τ/2 and τ . Because a backward path terminates in a

certain well with probability 1/n due to the symmetry of the potential and the initial

state of the time-reversed process, the probability that the backward path does not have

the corresponding forward path is

λsc =n− 1

n. (4.46)

Moreover, since we have no discrete path, we obtain

λd = 0. (4.47)


〈e−β(W−∆F )〉 =1

n. (4.48)

If we assume that K is sufficiently large, ∆F = 0 in this process. Thus, we obtain

〈e−βW 〉 =1

n. (4.49)

The corresponding inequality reads

〈βW 〉 ≥ lnn. (4.50)

We obtain the probability distributions of W at different n by numerical simulations

as shown in Fig. 4.4 (b). Based on this probability distribution, the value of 〈e−βW 〉 is

obtained and confirmed to be 1/n (see Fig. 4.4 (c)). We also verify that the average

dissipation 〈βW 〉 is larger than the minimum dissipation given by Eq. (4.50), namely, the

fundamental lower bound due to the absolute irreversibility as demonstrated in Fig. 4.4

(d). We note that this process may be regarded as an information erasure of an n-digit

memory.

4.3.3 System with a trap

Finally, we consider an overdamped Langevin system with a trap. The system is one-

dimensional, and the Langevin particle is confined in a single harmonic potential with

stiffness k(t). We assume that there is a trapping point in the system and the distance

between the center of the harmonic potential and the trapping point is denoted by xc

as illustrated in Fig. 4.5 (a). If the particle reaches the trapping point, it is trapped

78

with unit probability. Initially, the system is prepared in an equilibrium of the harmonic

potential. We subject the system to a nonequilibrium process by changing stiffness k.

The stiffness is decreased from k = K to 0 at a constant rate between t = 0 and τ/2, and

then increased from k = 0 to K at a constant rate between t = τ/2 to τ . Since the initial

probability can be written by the probability density, Eq. (4.39) reduces to

〈e−∆stot〉 = 1− λsc − λd. (4.51)

To evaluate the singular probabilities, we consider the time-reversed process. The

initial state of the time-reversed process is set to the final state of the original process.

Let ptrap and p†trap denote the trapping probabilities of the final state of the original and

time-reversed processes, respectively. Because the particle trapped in the original process

will remain trapped in the entire time-reversed process, the probability of this single path

is ptrap(> 0). Therefore, the discrete probability is

λd = ptrap. (4.52)

Moreover, since the paths that fall into the trap in the time-reversed process have no cor-

responding counterparts in the original process, they are singular continuous. Therefore,

the singular continuous probability is

λsc = p†trap − ptrap. (4.53)

Thus, Eq. (4.51) reduces to

〈e−∆stot〉 = 1− p†trap, (4.54)

which leads to an inequality

〈∆stot〉 ≥ − ln(1− p†trap). (4.55)

This inequality is automatically satisfied because the left-hand side is positively divergent

due to the presence of those paths that fall into the trap in the original path with a

positively divergent entropy production.

Figure 4.5 (b) shows the probability density of the total entropy production. Based on

this probability distribution, the exponentiated average of the total entropy production

is calculated and confirmed to be consistent with Eq. (4.54) as demonstrated in Fig. 4.5

(c).

79

(a) (b)

(c)

Figure 4.5: (a) Schematic illustration of a one-dimensional overdamped system witha trapping center. In the original process, the system is in an equilibrium of the har-monic potential. (b) Probability density of the total entropy production. Note thatthis probability density is not normalized over R because we have a positively diver-gent entropy production for the trapped paths. The triangles indicate the points with∆stot = − ln(1−p†trap). (c) Values of 〈e−∆stot〉 versus the trapping probability p†trap. Super-

imposed is the 1− p†trap line (with no free parameters). The parameters D and T are thesame as in Fig. 4.4. The distance between the center of the harmonic potential and thetrapping point is xc = 10−6m. The initial stiffness K is set so as to satisfy Ka2/2 = 10kBT .The duration of the process τ is varied between 1sec to 100sec to change the trappingprobability. The statistical average is taken over 106 samples for each τ . Reproducedfrom Fig. 5 of Ref. [27]. Copyright 2014 by the American Physical Society.

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/65羽田野研セミナー 2014年6月18日

自由膨張

80

対応する順経路なし

Figure 4.6: Virtual process corresponding to free expansion. Initially, the single-particlegas is in a global equilibrium state; the particle is in the left box with probability l andin the right box with probability 1− l. Then, the partition is removed. Reproduced fromFig. 8 of Ref. [27]. Copyright 2014 by the American Physical Society.

4.4 Comparison with a conventional method

In this section, we review a conventional method [64] to compare with our method based

on absolute irreversibility.

As an illustration, we rederive our nonequilibrium equality in free expansion, namely,

Eq. (4.40) by using the conventional method. We consider a virtual process starting from

a global equilibrium state with temperature T as shown in Fig. 4.6. Namely, the particle

is in the left box with probability l and in the right box with probability 1− l. Therefore,

the probability of paths starting from the right box does not vanish, and absolutely

irreversible paths do not exist. Thus, the conventional nonequilibrium (2.17) applies as

〈〈F [Γ]e−β(W−∆F )〉〉 = 〈〈F [Γ]〉〉†, (4.56)

where F [Γ] is an arbitrary functional, and 〈〈· · · 〉〉 denotes the statistical average of the

virtual process, and the caret ˆ means that the accompanying quantity is the one in the

virtual process. In this simple case, we can easily obtain relations between the physical

quantities in the virtual process and the corresponding quantities in the original process

81

as

W = W, (4.57)

F0 = F0 + kBT ln l, (4.58)

Fτ = Fτ , (4.59)

∆F = ∆F − kBT ln l. (4.60)

To derive a nonequilibrium equality in the original process, we set F [Γ] to the charac-

teristic functional whose value is equal to one only if the path starts from the left box

and zero otherwise. Then, considering the normalization of probabilities properly, we can

express the average in the original process in terms of that in the virtual process as

〈〈F [Γ] · · · 〉〉 = l〈· · · 〉. (4.61)

In the time-reversed process, the probability of paths ending in the right box is l, and

therefore we obtain

〈〈F [Γ]〉〉† = l. (4.62)


〈e−β(W−∆F )〉 = l, (4.63)

which agrees with Eq. (4.40) derived from the nonequilibrium equalities with absolute

irreversibility. Following a similar procedure, we can also rederive Eqs. (4.45) and (4.51)

in principle.

Let us examine the meaning of this conventional procedure. First, we extend the initial

probability distribution to the global canonical distribution of the system to circumvent

the problem of the vanishing probability. Then, we derive the nonequilibrium equality

(4.56) in this artificial process. Next, we express physical quantities of the artificial process

by those of the original process as in Eqs. (4.57) and (4.60). Finally, we erase some paths

irrelevant to the original dynamics by setting F to the characteristic function, and relate

the average of the virtual process to that of the original process as in Eq. (4.61).

In this way, the conventional method must introduce the artificial process to derive

the nonequilibrium equality to avoid the problem arising from absolute irreversibility. In

contrast, our method directly deals with the original process. Moreover, the reason why

the right-hand sides of Eqs. (4.40), (4.45), and (4.51) deviate from one is clearer in our

method, that is, the probability of absolutely irreversible paths should be subtracted from

82

one. Incidentally, we note that although Eq. (4.51) can in hindsight be derived by the

conventional method, we naturally find this absolutely irreversible example by virtue of

the stronger version of Lebesgue’s decomposition (4.13).

83

Chapter 5

Information-Thermodynamic

Nonequilibrium Equalities in

Absolutely Irreversible Processes

In this chapter, we generalize the nonequilibrium equalities in the presence of absolute ir-

reversibility obtained in Chap. 4 to situations under measurements and feedback control.

This generalization is of vital importance when error-free measurements are preformed,

because they project the probability distribution of the measured state onto a localized re-

gion in the phase space. The subsequent time evolution of this confined post-measurement

state, in general, involves expansion into an initially unoccupied region, which provides

yet another example of absolute irreversibility. The generalized nonequilibrium equalities

under measurements and feedback control enable us to identify the unavailable informa-

tion, which results from the inevitable loss of information that we cannot utilize under

a fixed feedback protocol. The unavailable information provides a fundamental limit of

performance of the feedback protocol.

First, we derive information-thermodynamic nonequilibrium equalities in the presence

of absolute irreversibility. Next, we introduce a notion of unavailable information and de-

rive a different type of nonequilibrium equalities that involve the unavailable information.

Finally, we demonstrate our nonequilibrium equalities in several examples.

This chapter is partly based on Refs. [27, 28].

5.1 Inforamtion-thermodynamic equalities

In this section, we derive information-thermodynamic nonequilibrium equalities in the

presence of absolute irreversibility. This section is mainly based on Ref. [27].

Under feedback control, it is important to consider the effect of the singular part of

84

the reference probability measure discussed in Sec. 4.1.2 because high-precision measure-

ments such as error-free measurements localize the probability distribution. Since the

feedback control starts from this post-measurement state confined in a narrow region, the

subsequent time evolution involves expansion into an initially unoccupied region unless

the feedback protocol is fine-tuned, and therefore the process exhibits absolute irreversibil-

ity. In experiments, since we have access to only a few parameters, the fine-tuning is a

difficult task in general. Hence, absolute irreversibility has experimental relevance in a

system under measurements and feedback control.

We consider a nonequilibrium process in a classical system with measurements and

feedback control from time t = ti to tf as in Sec. 3.3.2. Let x(t) and y(t) denote phase-

space points at time t of the system and the measurement outcomes, respectively. The

external control parameter λ(t) is determined based on the measurement outcome before

t, namely, y(s)ts=ti . Let X, Y and Λ[Y ] denote the entire paths of the system, the

measurement outcomes and the control parameter, respectively. Moreover, letM|Y denote

the conditional probability measure of X under given measurement outcomes Y , and let

Mr|Λ[Y ] denote the reference probability measure of X under a given feedback protocol

Λ[Y ]. We apply Lebesgue’s decomposition theorem to Mr|Λ[Y ] with respect to M|Y , and

obtain

Mr|Λ[Y ] =Mr

AC|Λ[Y ] +MrS|Λ[Y ], (5.1)

where the first and second terms on the right-hand side give the absolutely continuous and

singular parts of the reference probability measure, respectively. Lebesgue’s decomposi-

tion theorem ensures that this decomposition is unique. Let F [X, Y ] denote an arbitrary

path functional. It follows from Eq. (5.1) that

〈F〉r|Λ[Y ] = 〈F〉rAC|Λ[Y ] + 〈F〉rS|Λ[Y ], (5.2)

where 〈· · · 〉rI|Λ[Y ] (I = ∅,AC, S) denotes the average over MrI|Λ[Y ]. Moreover, the average

over the absolutely continuous part can be transformed by the Radon-Nikodym derivative

because

〈F〉rAC|Λ[Y ] =

∫F [X, Y ]Mr

AC|Λ[Y ][DX]

=

∫F [X, Y ]

DMrAC|Λ[Y ]

DM|Y

∣∣∣∣X

M|Y [DX]

= 〈Fe−R〉|Y , (5.3)

85

where 〈· · · 〉|Y denotes the average over M|Y , and we define

e−R =DMr

AC|Λ[Y ]

DM|Y

∣∣∣∣X

. (5.4)


〈Fe−R〉|Y = 〈F〉r|Λ[Y ] − 〈F〉rS|Λ[Y ]. (5.5)

If we set F to unity, we obtain

〈e−R〉|Y = 1− λS|Λ[Y ], (5.6)

where

λS|Λ[Y ] =

∫Mr

S|Λ[Y ][DX] (5.7)

is the probability of the singular part of the reference probability measure conditioned by

Λ[Y ].

Here, we consider the physical meaning of R defined in Eq. (5.4)1. The Crooks fluc-

tuation theorem (4.12) applies to the ratio of the transition probabilities under a fixed

protocol:

e−σ =DMr

AC|Λ[Y ]

DMtr|Λ[Y ]

∣∣∣∣∣X

, (5.8)

where Mtr|Λ[Y ] is the transition probability under the fixed protocol Λ[Y ]. Therefore,

R is different from σ because the conditional probability is different form the transition

probability as discussed in Sec. 3.2.2. The difference is represented by the Radon-Nikodym

derivative as

e−R = e−σDMtr

|Λ[Y ]

DM|Y

∣∣∣∣∣X

. (5.9)

Comparing this with Eq. (3.133), we find that the Radon-Nikodym derivative in this

equation may be written as e−I−Ii , where I is the total mutual information obtained by

1 Here, we assume that MrAC|Λ[Y ] is absolutely continuous with respect to Mtr

|Λ[Y ] and that Mtr|Λ[Y ]

is absolutely continuous with respect to M|Y . Otherwise, we cannot define the entropy productionand mutual information individually, because their values are divergent and mathematically ill-defined.However, the sum of these quantities remains finite due to the absolute continuity of Mr

AC|Λ[Y ] withrespect to M|Y .

86

the measurements and Ii is the initial correlation between the system and the measurement

outcomes. Therefore, we have

R = σ + I + Ii, (5.10)

and Eqs. (5.5) and (5.6) reduce to

〈Fe−σ−I−Ii〉|Y = 〈F〉r|Λ[Y ] − 〈F〉rS|Λ[Y ], (5.11)

〈e−σ−I−Ii〉|Y = 1− λS|Λ[Y ], (5.12)

respectively. Averaging these over measurement outcomes Y , we obtain

〈Fe−σ−I−Ii〉 = 〈F〉r − 〈F〉rS, (5.13)

〈e−σ−I−Ii〉 = 1− λS, (5.14)

where λS is the average over Y of the Λ[Y ]-conditioned singular probability λS|Λ[Y ]. Equa-

tion (5.14) derives a second-law-like inequality as

〈σ〉 ≥ −〈I + Ii〉 − ln(1− λS). (5.15)

Therefore, the lower bound of the entropy production is determined not only by the

mutual information but also by the term arising from the absolute irreversibility. If the

feedback protocol is so poorly designed that − ln(1 − λS) > 〈I + Ii〉, Eq. (5.15) implies

that the entropy production is positive and the feedback protocol does not work. In the

case of vanishing initial correlations, we obtain

〈Fe−σ−I〉 = 〈F〉r − 〈F〉rS, (5.16)

〈e−σ−I〉 = 1− λS, (5.17)

〈σ〉 ≥ −〈I〉 − ln(1− λS). (5.18)

Moreover, if the conditioned initial states M|Y satisfy the assumption of the stronger

version of Lebesgue’s decomposition (4.13), we obtain

〈Fe−σ−I〉 = 〈F〉r − 〈F〉rsc − 〈F〉rd, (5.19)

〈e−σ−I〉 = 1− λsc − λd, (5.20)

〈σ〉 ≥ −〈I〉 − ln(1− λsc − λd), (5.21)

where the subscripts “sc” and “d” represent the singular continuous and discrete parts,

87

respectively.

In the same manner as the case without feedback control, proper choices of the refer-

ence probability lead to nonequilibrium equalities with specific meanings of the entropy

production as listed in Table 4.2, and we obtain the corresponding equalities

〈e−β(W−∆F )−I〉 = 1− λS, (5.22)

〈e−∆stot/kB−I〉 = 1− λS, (5.23)

〈e−Ω−I〉 = 1− λS, (5.24)

〈e−∆shk/kB−I〉 = 1− λS, (5.25)

〈e−∆φ−∆sex/kB−I〉 = 1− λS, (5.26)

under the proper assumptions described in Sec. 2.2.

5.2 Unavailable information and associated equali-

ties

In this section, we define a concept of unavailable information based on the information-

thermodynamic nonequilibrium equalities obtained in the previous section, and derive new

nonequilibrium equalities that involve the unavailable information. Inequalities derived

from the new equalities give an achievable lower bound of the entropy production in the

case of error-free measurements.

First of all, we point out that the equality of Eq. (5.18) cannot be achieved in general

even in the quasi-static limit. The equality condition of Jensen’s inequality 〈e−x〉 ≥ e−〈x〉

is that the quantity x does not fluctuate. On the other hand, in the quasi-static limit,

σ is expected to have a single definite value. Therefore, in situations without feedback

control, the equality in the inequality

〈σ〉 ≥ − ln(1− λS) (5.27)

is achieved. In contrast, the equality in the inequality under feedback control

〈σ〉 ≥ −〈I〉 − ln(1− λS) (5.28)

cannot be achieved in general even in the quasi-static limit due to fluctuations of I, that is,

I depends on the measurement outcomes unless all outcomes are obtained with the same

probability. Hence, Eq. (5.28) gives only a loose lower bound of the entropy production,

although it is tighter than the conventional inequality (〈σ〉 ≥ −〈I〉) [18].

88

To find an achievable lower bound of the entropy production, we start from Eq. (5.12).

If we assume that we have no initial correlations, Eq. (5.12) reduces to

〈e−σ−I〉|Y = 1− λS|Λ[Y ]. (5.29)

From Jensen’s inequality, we obtain

〈σ〉|Y ≥ −〈I〉|Y − ln(1− λS|Λ[Y ]). (5.30)

What is noteworthy on this inequality is that the equality is achievable under error-free

measurements in the quasi-static limit, because I reduces to the unaveraged Shannon

entropy of Y , which has a single definite value under fixed measurement outcomes Y .

Therefore, the right-hand side of Eq. (5.30) gives an achievable lower bound of the entropy

production for a fixed Y . Hence, the last term in Eq. (5.30) represents the inevitable

dissipation due to the incompleteness of the feedback protocol Λ[Y ]. In other words,

the feedback protocol Λ[Y ] cannot fully utilize the mutual information obtained by the

measurements. Thus, we define unavailable information in the protocol Λ[Y ] as

Iu|Λ[Y ] = − ln(1− λS|Λ[Y ]) (≥ 0). (5.31)

We note that, under error-free measurements and in the quasi-static limit. we have

σ = −I + Iu (error-free, quasi-static). (5.32)

Using Eq. (5.31), we rewrite Eq. (5.29) as

〈e−σ−I+Iu〉|Y = 1. (5.33)

Averaging this equality over Y , we obtain a new nonequilibrium equality:

〈e−σ−I+Iu〉 = 1. (5.34)

This equality leads to

〈σ〉 ≥ −〈I − Iu〉. (5.35)

We note that this inequality is stronger than Eq. (5.28) due to the convexity of the

logarithmic function − lnx, that is,

〈Iu〉 = 〈− ln(1− λS|Λ[Y ])〉 ≥ − ln(1− λS). (5.36)

89

Furthermore, Eq. (5.35) gives an achievable lower bound of the entropy production under

error-free measurements and in the quasi-static limit because of Eq. (5.32). Therefore, we

can quantitatively characterize the incompleteness of the feedback protocol by calculating

the unavailable information.

Here, we assume that the initial state is an equilibrium state, and set the reference

probability to the time-reversed one starting from an equilibrium state. Then, Eqs. (5.34)

and (5.35) reduce to Jarzynski-type equations as

〈e−β(W−∆F )−I+Iu〉 = 1, (5.37)

−〈W 〉 ≤ −∆F + kBT 〈I − Iu〉, (5.38)

respectively. Equation (5.38) gives an achievable upper bound of extractable work in the

feedback process. The bound is reduced due to the unavailable information compared

with the conventional result [18].

The other choices of the reference probability summarized in Table 2.1 also give

nonequilibrium equalities with their individual meanings.

5.3 Examples of absolutely irreversible processes

In this section, we verify the information-thermodynamic nonequilibrium equalities de-

rived in the previous sections. Here, we only consider relations with dissipated work,

namely,

〈e−β(W−∆F )−I〉 = 1− λS, (5.39)

〈W 〉 −∆F ≥ −kBT 〈I〉 − kBT ln(1− λS), (5.40)

〈e−β(W−∆F )−I+Iu〉 = 1, (5.41)

〈W 〉 −∆F ≥ −kBT 〈I − Iu〉. (5.42)

First, we consider a measurement and subsequent trivial feedback control. Second,

we discuss the two-particle Szilard engine. Finally, the multi-particle Szilard engine is

considered.

5.3.1 Measurement and trivial feedback control

We consider a measurement and the subsequent feedback control. Initially, an ideal single-

particle gas is in a global equilibrium state of the box as illustrated in Fig. 5.1 (a). At

a certain time, we perform an instantaneous error-free measurement to determine the

90

/65羽田野研セミナー 2014年6月18日 4

(a)

X=RX=L

(b)

X=L X=RForward Backward

Figure 5.1: (a) A measurement and the subsequent trivial feedback control. Initially,an ideal single-particle gas is in a global equilibrium state of the entire box. At a certaintime, we perform an error-free position measurement to determine whether the particle isin the left or right side. After the measurement, we do nothing as the feedback control.(b) The time-reversed protocol. Initially, the particle is in the global equilibrium of theentire box and we do nothing regardless of the outcome of the forward process. At thetime of the measurement of the forward process, the particle is stochastically in the leftor right side. For the outcome X = L (R), the case in which the particle is in the right(left) side is singular (blue arrows). Reproduced from Fig. 6 of Ref. [27]. Copyright 2014by the American Physical Society.

position X of the particle. An outcome X = L is obtained when the particle is in the left

side, that is, the length from the left-end wall is shorter than the length of the whole box

multiplied by l (0 < l < 1). On the other hand, an outcome X = R is obtained when the

particle is in the right side. In both cases, we conduct trivial feedback control, i.e., we

leave the system as it is. Therefore, the gas expands to the entire box.

In this process, work is not extracted: W = 0, and the free energy does not change:

∆F = 0. Let p(L) and p(R) denote the probabilities to obtain outcomes L and R,

respectively. If the measurement outcome is X = L, we obtain mutual information I =

− ln p(L) = − ln l. If X = R, we obtain mutual information I = − ln p(R) = − ln(1− l).Next, we consider the time-reversed process to find the singular probabilities. In the

time-reversed process, we do nothing for both X = L and R, because we do nothing in the

forward process except for the measurement. The time-reversed process corresponding to

the outcome X = L is illustrated in the left half of Fig. 5.1 (b). The particle is in the

left side with probability l and in the right side with probability 1 − l. The case of the

particle being found in the right side has no corresponding forward event under X = L,

91

and therefore it is a singular event. Therefore, we obtain

λS|Λ[L] = 1− l. (5.43)

On the other hand, in the time-reversed process corresponding to the outcome X = R

illustrated in the right half of Fig. 5.1 (b), we obtain

λS|Λ[R] = l. (5.44)

Therefore, the unavailable information is obtained as

Iu|Λ[L] = − ln(1− λS|Λ[L]) = − ln l, (5.45)

Iu|Λ[R] = − ln(1− λS|Λ[R]) = − ln(1− l). (5.46)

We note that the unavailable information coincides with the mutual information in this

case, because all the information is lost since we do nothing as feedback control. The

values of the physical quantities are summarized in Table 5.1.

The left-hand side of Eq. (5.39) is calculated as

〈e−β(W−∆F )−I〉 = p(L) · eln l + p(R) · eln(1−l)

= l2 + (1− l)2. (5.47)

On the other hand, the averaged singular probability is

λS = p(L)λS|Λ[L] + p(R)λS|Λ[R]

= 2l(1− l). (5.48)

Therefore, by a simple calculation, we can verify Eq. (5.39). Next, let us verify Eq. (5.40).

The average of the dissipated work is

〈W 〉 −∆F = 0. (5.49)

Table 5.1: Values of physical quantities corresponding to the outcomes L and R.

X p(X) W ∆F I = − ln p(X) λS|Λ[X] Iu|Λ[X] = − ln(1− λS|Λ[X])

L l 0 0 − ln l 1− l − ln l

R 1− l 0 0 − ln(1− l) l − ln(1− l)

92

The right-hand side of Eq. (5.40) is

− kBT 〈I〉 − kBT ln(1− λS) = kBT [l ln l + (1− l) ln(1− l)− ln(l2 + (1− l)2)].(5.50)

By analytic calculation, we can show that the right-hand side of Eq. (5.50) is nonpositive,

that is,

kBT [l ln l + (1− l) ln(1− l)− ln(l2 + (1− l)2)] ≤ 0, (5.51)

and it is zero if and only if l = 1/2. Thus, Eq. (5.40) is verified. Moreover, the necessary

and sufficient condition for the equality is l = 1/2, which is consistent with the observation

made in Sec. 5.2, namely, the equality in the inequality (5.28) is achieved only when all

outcomes are obtained with the equal probability.

Now, we verify Eqs. (5.41) and (5.42) in the presence of the unavailable information.


〈e−β(W−∆F )−I+Iu〉 = p(L) · eln l−ln l + p(R) · eln(1−l)−ln(1−l) = 1, (5.52)

which verifies Eq. (5.41). Moreover, because Iu = I in this case, the right-hand side of

Eq. (5.42) is

− kBT 〈I − Iu〉 = 0. (5.53)


W −∆F = −kBT 〈I − Iu〉. (5.54)

This equality means that the equality in the inequality (5.42) is achieved regardless of the

value of l, which is what we expect because Eq. (5.42) gives an achievable bound.

5.3.2 Two-particle Szilard engine

Next, we consider the two-particle Szilard engine. The reason why we do not consider the

single-particle Szilard engine as in Refs. [20, 24] is that the singular part does not arise

in the single-particle Szilard engine since we can fully utilize the information obtained

by the measurements. Therefore, we should consider a Szilard engine with two or more

particles to observe effects of the absolute irreversibility. We begin by the two-particle

Szilard engine.

Initially, two identical ideal-gas particles are confined in a box as illustrated in Fig. 5.2.

93


2粒子Szilardエンジン

2

(b)

(a)

Figure 5.2: (a) Two-particle Szilard engine. Initially, an ideal two-particle gas is in aglobal equilibrium of the entire box. Then, we insert a partition in the middle of the boxand perform an error-free measurement to find the number n of the particles in the rightbox. Depending on the outcome, we isothermally and quasi-statically shift the partition tosuch a position that the extracted work is maximal. Finally, we remove the partition. (b)Time-reversed protocol for each n. The partition is inserted and then shifted in accordancewith the value of n. We have singular paths without the corresponding forward pathsin the case of n = 1 indicated by blue arrows. Reproduced from Fig. 7 of Ref. [27].Copyright 2014 by the American Physical Society.

94

We insert a partition in the middle of the box. Then, we perform an error-free measure-

ment to find the number n of the particles in the right side. If n = 0, we isothermally

and quasi-statically shift the wall to the right end to extract work, and then remove the

partition. If n = 2, we shift the wall to the opposite direction to obtain the same amount

of work, and then remove the partition. If n = 1, we just remove the wall because we

cannot extract work by shifting the partition. For all n, the system returns to the initial

state after these protocols. The values of extracted work are summarized in Table 5.2.

We consider the time-reversed process to obtain the singular probabilities (see Fig. 5.2).

Initially, the two particles are in a global equilibrium regardless of n. When n = 0, we

insert the partition in the right end of the box. Then, we isothermally and quasi-statically

shift the partition to the middle of the box and remove the partition. Because this path

has the counterpart in the forward process, we have no singularity when n = 0:

λS|Λ[0] = 0. (5.55)

Similarly, we have no singular part when n = 2:

λS|Λ[2] = 0. (5.56)

Now, let us consider the time-reversed process when n = 1. In this case, we insert the

partition in the middle of the box and then remove it. The probability of the two particles

being found in the left box has nonvanishing probability, so does the probability of both

particles being found in the right box. However, we have no corresponding forward paths

in the protocol of n = 1. Therefore, these time-reversed paths are singular, and the

singular probability is

λS|Λ[1] =1

2. (5.57)

Table 5.2: Values of physical quantities corresponding to the outcome n.

n p(n) W ∆F I = − ln p(n) λS|Λ[n] Iu|Λ[n] = − ln(1− λS|Λ[n])

01

4−2kBT ln 2 0 ln 4 0 0

11

20 0 ln 2

1

2ln 2

21

4−2kBT ln 2 0 ln 4 0 0

95

Thus, the unavailable information is

Iu|Λ[0] = 0, (5.58)

Iu|Λ[1] = ln 2, (5.59)

Iu|Λ[2] = 0. (5.60)

Now, we are ready to verify our nonequilibrium equalities. The left-hand side of

Eq. (5.39) is calculated as

〈e−β(W−∆F )−I〉 = p(0) · e2βkBT ln 2−ln 4 + p(1) · e− ln 2 + p(2) · e2βkBT ln 2−ln 4 =3

4. (5.61)

On the other hand, the average of the singular part is

λS = p(0)λS|Λ[0] + p(1)λS|Λ[1] + p(2)λS|Λ[2] =1

4. (5.62)

Therefore, Eq. (5.39) is verified. The left-hand side of Eq. (5.40) is

〈W 〉 −∆F = −kBT ln 2. (5.63)

On the other hand, the right-hand side of Eq. (5.40) is

− 〈I〉 − ln(1− λS) = − ln3√2. (5.64)

Hence, we obtain

〈W 〉 −∆F > −kBT 〈I〉 − kBT ln(1− λS), (5.65)

and Eq. (5.40) is verified. We note that the equality in the inequality (5.40) is not achieved

because the probability of finding a particular value of n varies for each n.


〈e−β(W−∆F )−I+Iu〉 = p(0) · e2βkBT ln 2−ln 4 + p(1) · e− ln 2+ln 2 + p(2) · e2βkBT ln 2−ln 4 = 1, (5.66)

which verifies Eq. (5.41). The right-hand side of Eq. (5.42) is

− kBT 〈I − Iu〉 = −kBT ln 2. (5.67)

96

Thus, we obtain

〈W 〉 −∆F = −kBT 〈I − Iu〉, (5.68)

verifying Eq. (5.42). Because Eq. (5.42) gives an achievable bound of work in the quasi-

static limit, the equality in Eq. (5.42) is achieved.

5.3.3 Multi-particle Szilard engine

We generalize the two-particle Szilard engine to the multi-particle case. Initially, an ideal

N -particle gas is in a global equilibrium of the entire box. We insert a partition in the

middle of the box. Then, we measure the number n of the particles in the right box.

Then, we isothermally and quasi-statically shift the partition to the position that divides

the entire box according to the ratio (N − n) : n. Finally, we remove the partition. The

probability of the outcome n being found is

p(n) =NCn2N

, (5.69)

where NCn represents the number of combination that we choose n objects from N objects.

Therefore, the information obtained by the measurement is

I(n) = N ln 2− lnNCn. (5.70)

The work during the process can be calculated as

W = −nkBT ln2n

N− (N − n)kBT ln

2(N − n)

N

= −nkBT lnn

N− (N − n)kBT ln

(N − n)

N−NkBT ln 2. (5.71)

Since the system returns to the initial state, we have

∆F = 0. (5.72)

We calculate the singular probabilities by considering the time-reversed process, and ob-

tain

λS|Λ[n] = 1− NCn

( nN

)n(N − nN

)N−n, (5.73)

97

and

Iu|Λ[n] = − lnNCn − n lnn

N− (N − n) ln

N − nN

. (5.74)

The left-hand side of Eq. (5.39) is

〈e−β(W−∆F )−I〉 =N∑n=0

p(n) exp

[n ln

n

N+ (N − n) ln

N − nN

+ lnNCn

]

=N∑n=0

p(n)( nN

)n(N − nN

)N−nNCn

=N∑n=0

(NCn)2

2N

( nN

)n(N − nN

)N−n. (5.75)

On the other hand, the averaged singular probability is

λS =N∑n=0

p(n)λS|Λ[n] (5.76)

=N∑n=0

p(n)

[1− NCn

( nN

)n(N − nN

)N−n](5.77)

= 1−N∑n=0

(NCn)2

2N

( nN

)n(N − nN

)N−n. (5.78)

Therefore, Eq. (5.39) is verified.

By a simple calculation, we obtain

W −∆F = −kBT (I − Iu). (5.79)

Hence, Eqs. (5.41) and (5.42) are automatically satisfied. Moreover, the equality in

Eq. (5.42) is achieved, because the measurement is error-free and the process is quasi-

static.

Large-N limit

We consider the large-N limit of the N -particle Szilard engine. The fluctuation of the

particle number n in the right box after the partition is inserted is of the order of√N .

Therefore, the dominant contributions to the average of physical quantities come from

events with

n =N

2(1 + x), x = O

(1√N

). (5.80)

98

We consider approximate formulae in this range of n.

The logarithm of the probability p(n) is calculated as

ln p(n) = lnN !− lnn!− ln(N − n)!−N ln 2. (5.81)

By using the Stirling formula

lnN ! =1

2ln(2πN) +N ln

N

e+O

(1

N

), (5.82)

we obtain

ln p(n) =1

2ln

N

2πn(N − n)+N lnN − n lnn− (N − n) ln(N − n)−N ln 2 +O

(1

N

)=

1

2ln

2

πN(1− x2)− N

2(1 + x) ln(1 + x)− N

2(1− x) ln(1− x) +O

(1

N

)=

1

2ln

2

πN− N

2x2 +O

(1

N

). (5.83)


p(n) =

√2

πNexp

[−N

2x2 +O

(1

N

)](5.84)

and

I(n) =1

2lnπN

2+N

2x2 +O

(1

N

). (5.85)

In a similar manner, the unavailable information is calculated as

Iu|Λ[n] = −1

2ln

N

2πn(N − n)+O

(1

N

)=

1

2lnπN

2+O

(1

N

). (5.86)

The average of the mutual information is evaluated as

〈I〉 =N∑n=0

p(n)I(n)

=N

2

∫ ∞−∞

p(n)I(n)dx

=1

2lnπN

2+

1

2+O

(1

N

). (5.87)

99

Therefore, the conventional second law of information thermodynamics

〈W 〉 −∆F ≥ −kBT 〈I〉 (5.88)

reveals that the effect of feedback control is sub-extensive and of the order of lnN .

On the other hand, the average of the unavailable information is obtained as

〈Iu〉 =1

2lnπN

2+O

(1

N

). (5.89)

Therefore, the amount of the available information is

〈I − Iu〉 =1

2+O

(1

N

). (5.90)

Thus, our inequality

〈W 〉 −∆F ≥ −kBT 〈I − Iu〉 (5.91)

limits the effect of feedback control to the order of unity, which is a qualitatively new

restriction compared with that of the conventional second law of information thermody-

namics. The same behavior was derived by another method [65] based on the Kawai-

Parrondo-Van den Broeck equality [66, 67].

100

Chapter 6

Gibbs’ Paradox Viewed from

Absolute Irreversibility

In this chapter, we apply our nonequilibrium equalities in the presence of absolute irre-

versibility to the problem of gas mixing. This problem is what Gibbs’ paradox concerns

[29]. Gibbs’ paradox is qualitatively resolved once we realize that there are many entropies

[30–32]. The most prevalent quantitative resolution based on quantum mechanics is in

fact irrelevant to Gibbs’ paradox [31, 32]. Another well recognized resolution is based

on the extensivity of the thermodynamic entropy [33, 32]. However, this resolution is

valid only in the thermodynamic limit, and cannot deal with any sub-leading effects. We

propose a new quantitative resolution of Gibbs’ paradox based on our nonequilibrium

equalities.

First of all, we review the original Gibbs’ paradox and later qualitative discussions.

Then, we present two widespread quantitative resolutions. Finally, we resolve Gibbs’

paradox from the viewpoint of absolute irreversibility.

6.1 History and Resolutions

In this section, we review a brief history of Gibbs’ paradox and discuss its resolutions.

We first review the original discussion and interpretation given by Gibbs. Next, we

review works by later researchers that clarify Gibbs’ interpretation and argue that Gibbs’

paradox is qualitatively resolved even in classical theory. Then, we discuss the quantum

resolution of Gibbs’ paradox, which is the standard resolution of Gibbs’ paradox. Finally,

we examine yet another resolution given by Pauli.

101

6.1.1 Original Gibbs’ paradox

First of all, we review Gibbs’ original discussion which appeared in his writing titled “On

the Equilibrium of Heterogeneous Substances” [29]. We consider mixing of two different

gases (see Fig. 6.1 (a)). The gases are confined in a box partitioned into equal halves.

Initially, an ideal N -particle gas of one kind is confined in the left side with temperature T ,

and an idealN -particle gas of the other kind is in the right side with the same temperature.

Then, we remove the partition and the two gases expand to the entire box. Let S(T, V,N)

denote the thermodynamic entropy of an ideal gas with temperature T , volume V and the

number of particles N . The entropy production of this mixing process can be calculated

by the Clausius definition of the thermodynamic entropy. The entropy production of one

gas is given by

S(T, 2V,N)− S(T, V,N) = −∫ (T,2V,N)

(T,V,N)

δQ

T, (6.1)

where Q denotes the heat transferred from the system to a heat bath, and the integral is

conducted along an arbitrary virtual quasi-static process that connects the state (T, V,N)

to (T, 2V,N). We set the virtual process to the quasi-static isothermal process. Due to

the first law of thermodynamics and the fact that the internal energy of an ideal gas does

not change in an isothermal process, we have

δQ = −pdV, (6.2)

where p is the pressure of the gas. Therefore, we obtain

S(T, 2V,N)− S(T, V,N) =

∫ 2V

V

pdV

T. (6.3)

Using the equation of state pV = NkBT , we obtain

S(T, 2V,N)− S(T, V,N) = NkB

∫ 2V

V

dV

V= NkB ln 2. (6.4)

Due to the additivity of the thermodynamic entropy, the entropy production of two dif-

ferent gases is the sum of their individual entropy productions. Therefore, the entropy

production of the mixing of two different gases is given by

∆Sdif = 2NkB ln 2. (6.5)

After Gibbs derived Eq. (6.5), he argued [29]

102

(a) (b)

Figure 6.1: (a) Mixing of two different gases. Initially, an ideal N -particle gas is confinedin the left side and another ideal N -particle gas of the other kind is confined in the rightside. Then, we remove the partition, and the two gases expand to the entire box. (b)Mixing of two identical gases. Initially, two ideal N -particle gases of the same kind areconfined both in the left and right sides. Then, we remove the partition, and the gasesexpand to the entire box.

“It is noticeable that the value of this expression does not depend upon the

kinds of gas which are concerned, if the quantities are such as has been sup-

posed, except that the gases which are mixed must be of different kinds. If

we should bring into contact two masses of the same kind of gas, they would

also mix, but there would be no increase of entropy.”

In fact, when we consider mixing of two identical gases (see Fig. 6.1 (b)), the initial

entropy of the system is given by

2S(T, V,N) (6.6)

due to the additivity of the thermodynamic entropy. On the other hand, the final entropy

is

S(T, 2V, 2N), (6.7)

which is equal to the initial entropy (6.6) due to the extensivity of the thermodynamic

entropy:

S(T, qV, qN) = qS(T, V,N), (6.8)

where q is an arbitrary positive real number. Therefore, the entropy production of the

103

mixing of two identical gases is

∆Sid = 0, (6.9)

which is different from the entropy production of the mixing of two different gases (6.5),

although Eq. (6.5) is “independent of the degrees of similarity or dissimilarity between

them.” In particular, if we consider

“the case of two gases which should be absolutely identical in all the proper-

ties (sensible and molecular) which come into play while they exist as gases

whether pure or mixed with each other, but which should differ in respect to

the attractions between their atoms and the atoms of some other substances,”

the thermodynamic entropy increases in the mixing process of two different gases, although

“the process of mixture, dynamically considered, might be absolutely identical

in its minutest details (even with respect to the precise path of each atom)

with processes which might take place without any increase of entropy,”

i.e., with processes of the mixing of identical gases. This paradoxical consequence of the

gas mixing is referred to as Gibbs’ paradox.

To explain this fact, Gibbs stressed,

“if we ask what changes in external bodies are necessary to bring the system

to its original state, we do not mean a state which shall be undistinguishable

from the previous one in its sensible properties. It is to states of systems thus

incompletely defined that the problems of thermodynamic relate.”

In other words, the thermodynamic entropy is not a property of a microstate, but a

property of a set of microstates indistinguishable from each other, i.e., a property of a

macrostate. In this way, “entropy stands strongly contrasted with energy.” Therefore,

“the mixture of gas-masses of the same kind stands on a different footing from

the mixture of gas-masses of different kinds,”

This explanation given by Gibbs was clarified by later researchers as we see in the next

section.

6.1.2 Gibbs’ paradox is not a paradox

In this section, we review discussions by three researchers, and argue that Gibbs’ paradox

is not a paradox even within classical statistical mechanics.

104

Grad [30]

Grad discussed Gibbs’ paradox in the introduction of his article titled “The Many Faces

of Entropy” [30]. In the introduction, he emphasized,

“A given object of study cannot always be assigned a unique value, its ‘en-

tropy.’ It may have many different entropies, each one worthwhile.”

and argued,

“much of the confusion in the subject is traceable to the ostensibly unifying

belief (possibly theological in origin!) that there is only one entropy.”

In fact, Gibbs’ paradox is resolved once we renounce this belief. Grad continued,

“Whether or not diffusion occurs when a barrier is removed depends not on a

difference in physical properties of the two substances but on a decision that

we are or are not interested in such a difference (which is what governs the

choice of an entropy function). There is no paradox to any observer. When

he is aware of a difference in properties, he observes diffusion together with

an increase in entropy. When he is unaware of any difference, he observes no

diffusion and no increase in the entropy which he is using. If two observers

disagree, they must be interested in different phenomena, and there is no

conflict.”

To clarify his idea, Grad raised different situations that describe states of an n-particle

gas. In the first situation, we label particles in the gas by 1, · · · , n. In the second situation,

we count the particle number of the gas in a certain region of space. These two situations

are different and give rise to two different non-comparable entropies. In the first situation,

when we remove the partition and mix the gas, the entropy S1 increases by n ln 2. Then,

when we reinsert the partition, the S1 decreases by n ln 2 and returns to its original value.

This is because we have complete information whether each particle is in the left or

right side since we label all the particles. In the second situation, when we remove the

partition, the entropy S2 increases by n ln 2, if we distinguish the particles originally in

the two different sides when we count the number of particles. On the other hand, S2

does not increase, if we are not interested in any difference of the particles. In both cases,

when we reinsert the partition, the entropy does not change. In this way, the entropies

in these descriptions differ from each other. Although S1 is completely impractical from

the viewpoint of thermodynamics, it does exist and give one description of the system.

In summary, the notion of entropy depends on our interest or description of a system.

If entropies differ from each other, it just implies a difference of our interest, and there

105

is no conflict. As to Gibbs’ paradox, the difference of the entropy production reflects our

situation of whether or not we are interested in a difference of the particles.

van Kampen [31]

In his essay, van Kampen discussed Gibbs’ paradox. The aim of his essay is to refute such

statements as

“It is not possible to understand classically why we must divide N ! to obtain

the correct counting of states,” and “Classical statistics thus leads to a con-

tradiction with experience even in the range in which quantum effects in the

proper sense can be completely neglected.”

First, he calculated the entropy S(P, T ) of an ideal N -particle gas with pressure P

and temperature T , and obtained

S(P, T ) =5

2NkB lnT −NkB lnP + C. (6.10)

Then, he emphasized,

“the second law defines only entropy differences between states that can be

connected by reversible change.”

Therefore, the constant C is only required to be independent of P and T . Thus, at this

point, there is no way to compare entropy with different N , unless we introduce a new

reversible process that varies N . We consider a process in which we attach two boxes

containing identical gases with the same (P, T,N), and open a channel between them.

Then, the constant C should be proportional to N and we obtain

S(P, T,N) =5

2NkB lnT −NkB lnP + cN, (6.11)

where c does not depend on (P, T,N). If two boxes contain two different gases A, B, the

process in which the channel is opened is no longer reversible. Instead, we need to consider

a process with semi-permeable walls. We combine two boxes using the semi-permeable

walls in a reversible way, and then isothermally and quasi-statistically expand the box

so that the pressure returns to its original value. Together with the convention of the

additivity of the thermodynamic entropy, we obtain

S(P, T,NA = N/2, NB = N/2) =5

2NkB lnT −NkB ln

P

2+N

2(cA + cB). (6.12)

Even when we set A=B, Eq. (6.12) does not reduce to Eq. (6.11). This difference is what

Gibbs’ paradox concerns. Then, van Kampen argued,

106

“The origin of the difference is that two different process had to be chosen for

extending the definition of entropy. They are mutually exclusive: the first one

cannot be used for two different gases and the second one does not apply to a

single one.”

Next, he considered the case in which A and B are so similar that an experimenter cannot

distinguish them operationally, namely, he does not have the semi-permeable walls needed

in the second process. Then, he will conclude

S(P, T,NA = N/2, NB = N/2) =5

2NkB lnT −NkB lnP + cN. (6.13)

This seems to be contradictory at a first glance, but

“The point is, that this is perfectly justified and that he will not be led to

any wrong results. If you tell him that ‘actually’ the entropy increased when

he opened the channel he will answer that this is a useless statement since

he cannot utilize the entropy increase for running a machine. The entropy

increase is no more physical to him than the one that could be manufactured

by taking a single gas and mentally tagging the molecules A or B. [...] The

expression for the entropy (which he constructs by one or the other processes

mentioned above) depends on whether or not he is able and willing to distin-

guish between the molecules A and B. This is a paradox only for those who

attach more physical reality to the entropy than is implied by its definition.”

Van Kampen concluded

“The question is not whether they are identical in the eye of God, but merely

in the eye of the beholder.”

Jaynes [32]

Jaynes argued that the writing of Gibbs [29] contains a correct analysis of Gibbs’ paradox.

However, this analysis has been lost due to ambiguity in the writing and the fact that

Gibbs did not include it in his later renowned textbook [68]. Jaynes presented a “half

direct quotation” of Gibbs’ explanation [32].

First of all, Jaynes stressed that we have to be circumspect about what we mean by

the words “state” and “reversible.” He wrote

“But by the word ‘original state’ we do not mean that every molecule has been

returned to its original position, but only a state which is indistinguishable

from the original one in the macroscopic properties that we are observing.”

107

Therefore, in the mixing of two different gases, the particles originally in the left (right)

side must return to the left (right) when we say that the state returns to the original

state. In contrast, in the mixing of two identical gases, we do not mean that the particles

originally in the left (right) side should return to the left (right) when we say that the

mixing is reversible and accompanied by no entropy production. We say that the mixing is

reversible because all macroscopic properties (e.g. the chemical composition, the number

of particles) return to their original value after we reinsert the partition. Thus,

“Trying to interpret the phenomenon as a discontinuous change in the physical

nature of the gases (i.e. in the behavior of their microstates) when they become

exactly the same, misses the point. [...] We might put it thus: when the gases

become exactly the same, the discontinuity is in what you and I mean by the

words ‘restore’ and ‘reversible’.”

To clarify this point, Jaynes continued his discussion. He noted that a thermodynamic

state is defined by specifying a small number of macroscopic quantities X1, X2, · · · , Xn.The entropy is defined as a property of a macrostate specified by these quantities: S =

S(X1, X2, · · · , Xn). As indicated by the discussion of the gas mixing, the entropy is not a

property of the microstate, whereas other thermodynamic variables as the total mass and

the total energy are physically real properties of a microstate. In fact, the thermodynamic

entropy is a property of a macrostate C(X), namely, a set of microstates compatible with

X = X1, X2, · · · , Xn. Thus, it is possible to assign different entropies S, S ′ to the same

microstate, if we choose different sets of macroscopic variables and embed the microstate

in two different macrostates C,C ′. This implies that we have to specify macroscopic vari-

ables that we can measure and control in advance to define the thermodynamic entropy.

Then, the thermodynamic entropy obeys the second law of thermodynamics as long as

all experimental manipulations are within the set of macroscopic variables that we have

chosen beforehand. Because this choice connotes whether we regard the gases as different

or identical, the behavior of the entropy under the gas mixing hinges upon this choice.

Summary

As we have seen, the thermodynamic entropy is not an intrinsic property of a microstate,

and the definition of the thermodynamic entropy involves arbitrariness. Whether the

gases are identical or different is predetermined within our thermodynamic framework

that we have chosen beforehand. Thus, it is meaningless to discuss the discontinuity in

the thermodynamic entropy when we consider the infinitely similar gases. In this sense,

Gibbs’ paradox is not a paradox.

108

Now that we qualitatively understand that Gibbs’ paradox is not a paradox even within

classical thermodynamics, the remaining question is how to derive the factorial in the sta-

tistical definition of the thermodynamic entropy, namely, the factor N ! in the definition

of the entropy

S = lnW

N !, (6.14)

where W denotes the number of microstates, or in the definition of the partition function

Z =1

N !

∫e−βH(Γ)dΓ, (6.15)

where H(Γ) is the Hamiltonian of the system. In the following two sections, we review

two well-known quantitative resolutions of Gibbs’ paradox.

6.1.3 Quantum resolution

The standard resolution of Gibbs’ paradox is based on quantum mechanics. In quantum

theory, the interchange of identical particles does not lead to another state, and identical

particles are indistinguishable in principle. This indistinguishability naturally leads to

the factor N !. Many textbooks (for example, see Refs. [69–73]) resolve Gibbs’ paradox in

this way.

Following Ref. [69], we consider an N -particle quantum system with a Hamiltonian

H =∑i

p2i

2m+ V (x1, · · · ,xN), (6.16)

where m is the mass of the particles. Let |i〉 and Ei denote the i-th eigenstate and eigenen-

ergy, respectively. Then, the (unnormalized) density matrix of the canonical ensemble is

ρβ =∑i

e−βEi|i〉〈i|. (6.17)

The configurational representation reads

ρβ(x1, · · · ,xN ; x′1, · · · ,x′N) =∑i

e−βEiψi(x1, · · · ,xN)ψ∗i (x′1, · · · ,x′N), (6.18)

where ψi(x1, · · · ,xN) = 〈x1, · · · ,xN |i〉. When the particles are bosons, this density

109

matrix must be symmetrized as

ρsymβ (x1, · · · ,xN ; x′1, · · · ,x′N) =

1

N !

∑σ∈SN

ρβ(x1, · · · ,xN ; x′σ(1), · · · ,x′σ(N)), (6.19)

where SN is the symmetry group of degree n. The partition function is given by

Zsymβ =

∫dx1 · · · dxN ρsym

β (x1, · · · ,xN ; x1, · · · ,xN)

=1

N !

∑σ∈SN

∫dx1 · · · dxN ρβ(x1, · · · ,xN ; xσ(1), · · · ,xσ(N)) (6.20)

The integrand in this equation involves factors

exp

[−m(xi − xσ(i))

2

2β~2

]. (6.21)

In the classical (i.e., high-temperature) limit, due to this exponential decay, the dominant

contribution in Eq. (6.20) is the term in which σ is the identity permutation. Therefore,

Eq. (6.20) is approximated as

Zsymβ ' 1

N !

∫dx1 · · · dxN ρβ(x1, · · · ,xN ; x1, · · · ,xN). (6.22)

In the case of fermions, a similar discussion leads to the same conclusion. In this way, the

factor N ! can naturally be derived from quantum mechanics and this factor leads to the

extensive thermodynamic entropy.

Although this resolution is the standard resolution of Gibbs’ paradox, it involves two

crucial problems. First, this resolution cannot apply to mesoscopic particles. Let us

consider colloidal particles in liquid. Because colloidal particles have vast internal degrees

of freedom, we cannot expect that these particles have the same internal states. Therefore,

the wave function of this system should not be symmetrized. Hence, we cannot derive the

factor N !, and thermodynamic quantities (e.g. entropy) fail to be extensive. This implies

that the quantum resolution is inapplicable to a mesoscopic regime. The second point is

more fundamental and to be explained at the end of the next section, because this point

is closely linked to the topic described in the next section.

6.1.4 Pauli’s resolution based on the extensivity

As we see in the review of van Kampen’s work [31], the Clausius definition of thermo-

dynamic entropy reveals nothing about the dependence of the entropy on the particle

number. Pauli recognized this fact and gave a resolution within classical theory [33, 32].

110

Pauli’s analysis is based on the extensivity of the thermodynamic entropy.

Let S(T, V,N) be a phenomenological thermodynamic entropy of an ideal N -particle

gas with temperature T and volume V defined by the Clausius definition (6.1). Then,

from the definition, we obtain

S(T, V,N) =3

2NkB lnT +NkB lnV + kBf(N), (6.23)

where f(N) is not an arbitrary constant, but an arbitrary function of N . This is because

the Clausius definition does not involve the N -dependence of the thermodynamic entropy.

To determine f(N), we require the extensivity of the entropy as an additional condition.

The extensivity means

S(T, qV, qN) = qS(T, V,N), (6.24)

where q is an arbitrary positive real number. Substituting Eq. (6.23) into Eq. (6.24), we

obtain

qN ln q + f(qN) = qf(N). (6.25)

Differentiating with respect to q and setting q = 1, we obtain

N +Nf ′(N) = f(N). (6.26)

This equation can be rewritten as

d

dN

(f(N)

N

)= − 1

N. (6.27)


f(N) = Nf(1)−N lnN, (6.28)

where the second term is the factor N ! due to the Stirling formula in the large-N limit,

i.e., in the thermodynamic limit. Thus, the thermodynamic entropy is

S(T, V,N) =3

2NkB lnT +NkB ln

V

N+NkBf(1). (6.29)

By this form, we see that the entropy is extensive and f(1) is essentially the chemical

potential. In this way, the requirement of the extensivity leads to the factor N ! in the

phenomenological entropy defined by the Clausius in the thermodynamic limit.

111

The same argument applies to the entropy defined by classical statistical mechanics.

The reason why we identify the entropy defined by classical statistical mechanics as the

thermodynamic entropy is that these two entropies have the same response to any vari-

ation of such macroscopic variables as the temperature and volume. In mathematical

terms, these two entropies have the identical differential form. Therefore, what we can

conclude about relations between the thermodynamic entropy S(T, V,N) and the entropy

in classical statistical mechanics SC(T, V,N) is

S(T, V,N) = SC(T, V,N) + kBfC(N). (6.30)

For an N -particle ideal gas, we have

SC(T, V,N) =3

2NkB[ln(2πmkBT ) + 1] +NkB lnV − 3NkB ln ξ, (6.31)

where ξ is a constant of the dimension of action. Then, the requirement of the extensiv-

ity (6.24) leads to

fC(N) = NfC(1)−N lnN. (6.32)

The extensivity reproduces the second term on the right-hand side, i.e., the factor N !

again.

Quantum statistical mechanics is in the the same position as classical statistical me-

chanics. We have

S(T, V,N) = SQ(T, V,N) + kBfQ(N). (6.33)

In the classical limit, since we have

SQ(T, V,N) ' SC(T, V,N)− lnN !, (6.34)

the requirement of the extensivity leads to

fQ(N) = NfQ(1) + lnN !

NN, (6.35)

where the second term on the right-hand side vanishes in the thermodynamic limit. Hence,

fQ(N) has only a trivial dependence on N as

fQ(N) ' NfQ(1), (6.36)

112

which corresponds to the contribution from the chemical potential. In this manner, the

procedure to determine an arbitrary function of N is needed for quantum statistical me-

chanics as well, although the result is simpler than that of classical statistical mechanics.

Therefore, [31]

“the Gibbs paradox is no different in quantum mechanics, it is only less man-

ifest.”

As we have seen, the quantum resolution is irrelevant to Gibbs’ paradox. The reso-

lution based on the extensivity is a better and logical resolution, and applicable to the

phenomenological entropy, the entropy in classical statistical mechanics and the entropy

in quantum statistical mechanics. However, this resolution still suffers a problem. The

resolution is only applicable to systems in the thermodynamic limit in which we are en-

titled to require the extensivity of the entropy. Namely, it ignores deviations from the

extensivity, which are essential to deal with mesoscopic physics and surface effects.

6.2 Resolution from absolute irreversibility

In this section, we resolve Gibbs’ paradox based on the nonequilibrium equalities with

absolute irreversibility. We explain why the entropy productions of the two mixing pro-

cesses are different from each other, and then derive the factor N !. Our resolution is valid

even for non-extensive entropy, for which the resolution given by Pauli breaks down.

6.2.1 Requirement and Results

In the resolution by Pauli, we require the extensivity of the thermodynamic entropy.

Instead of this requirement, we require the additivity of the thermodynamic entropy. In

mesoscopic systems, the extensivity breaks down, whereas the additivity remains valid

as long as the interaction is short-range. The additivity plays a crucial role when we

compare the thermodynamic entropy with different N .

Under the requirement of the additivity, in the context of our nonequilibrium equality,

we show that the entropy production of the mixing of two identical N -particle gases is

∆Sid = 0 (6.37)

and that the entropy production of the mixing of two different N -particle gases is

∆Sdif = 2NkB ln 2 (6.38)

113

in the thermodynamic limit. This difference of the two processes will be explained in terms

of absolute irreversibility. Moreover, we will derive the factor N !, that is, we show that

the arbitrary function fC(N) in classical statistical mechanics should take the following

form:

fC(N) = NfC(1)− lnN !. (6.39)

This result is valid for a finite N without the thermodynamic limit.

6.2.2 Difference of the two processes

We consider a difference between the mixing of two identical gases and that of two dif-

ferent gases in terms of absolute irreversibility. First, we consider the reverse process

of the mixing of two identical gases illustrated in Fig. 6.2 (a) to evaluate the absolute

irreversibility. Initially, an ideal 2N -particle gas is in the equilibrium of the entire box.

Then, we insert the partition in the middle. Let n denote the number of the particles

found in the left side after the insertion. The event of n = N is the only event that has the

corresponding event in the original process (see also Fig. 6.1 (b)). Therefore, the events

of n 6= N are singular because they have no counterparts. Hence, the singular probability

is calculated as

λidS = 1− 2NCN

22N. (6.40)

Secondly, we consider the reverse process of the mixing of two different gases illustrated

in Fig. 6.2 (b). Initially, two ideal N -particle gases of different kinds are at thermal

equilibrium in the entire box. Then, we insert the partition in the middle. To recover the

original state, the particle number in the left side after the insertion must be N . Moreover,

the chemical composition must return to the original state. Namely, the particles from

the left (right) side in the original process must return to the left (right) side. This fact

makes sharp contrast to the case of two identical gases, in which the particles from the

left (right) may go to the right (left) sides as long as the particle number returns to N .

The only non-singular event is the one in which all the particles of one kind return to the

left side and the rest particles return to the right side. All the other events indicated by

blue arrows are singular. Thus, the singular probability is

λdifS = 1− 1

22N. (6.41)

114

· · · · · ·(a)

(b)

· · ·

· · ·· · ·

· · ·

Figure 6.2: (a) Reverse process of the mixing of two identical gases. Initially, an ideal2N -particle gas is at thermal equilibrium in the entire box. Then, we insert the partitionin the middle. The particle number in the left side n varies from 0 to 2N according tothe binomial distribution. The events of n 6= N indicated by the blue arrows are singularbecause they have no counterparts in the original (i.e., forward) process. (b) Reverseprocess of the mixing of two different gases. Initially, two ideal N -particle gases of differentkinds are at thermal equilibrium of the entire box. Then, we insert the partition in themiddle. To return to the original state, not only must the particle number in the left sidebe N , but also the composition of gas must be the same as the original state. Therefore,the mixing of two different gases has much more absolute irreversibility than the mixingof two identical gases.

115

In this way, the difference of the intuitive physical descriptions in the reversed processes

is quantitatively characterized by the difference of the probabilities of the absolutely

irreversible paths.

Next, we connect these singular probabilities to the thermodynamic entropy produc-

tion. The Jarzynski-type nonequilibrium equality (4.33) in the presence of absolute irre-

versibility reads

〈e−β(W−∆F )〉 = 1− λS. (6.42)

Since work is zero (W = 0) in the mixing process, we obtain

∆F = kBT ln(1− λS). (6.43)

Thus, in terms of the thermodynamic entropy, we obtain

∆S = −kB ln(1− λS). (6.44)

Substituting Eq. (6.40) into Eq. (6.44), we obtain

∆Sid = 2NkB ln 2− kB ln 2NCN . (6.45)

When N is sufficiently large, the combination in Eq. (6.45) is approximated as

2NCN '22N

√πN

. (6.46)


∆Sid ' 1

2kB ln πN. (6.47)

Since this value is sub-extensive, we have

∆Sid = 0 (6.48)

in the thermodynamic limit. This is consistent with the fact that the removal of the

partition becomes reversible in the large-N limit due to the law of large numbers, namely,

the reinsertion of the partition results in the state of n = N with almost unit probability.

In the case of two different gases, from Eqs. (6.41) and (6.44), we obtain

∆Sdif = 2NkB ln 2. (6.49)

116

· · · · · ·(a) (b)

Figure 6.3: (a) Mixing of an N -particle gas and an M -particle gas of the same kind. Abox is divided into equal halves by a partition. Initially, the N -particle gas is in the leftside, and the M -particle gas is in the right side. We remove the partition and the gasesexpand to the entire box. (b) The reverse process. Initially, an (N +M)-particle gas is atthermal equilibrium in the entire box. Then, we insert the partition. The number of theparticles in the left side n varies from 0 to N +M according to the binomial distribution.

The difference of these entropy productions of the two processes originates from the dif-

ference of the degree of absolute irreversibility, namely, the difference in the behaviors

under the reinsertion of the partition.

6.2.3 Derivation of the factor N !

To determine the arbitrary function fC(N) in Eq. (6.30), we consider a mixing process

of an N -particle gas and an M -particle gas of the same kind illustrated in Fig. 6.3 (a).

To evaluate the probability of absolute irreversibility, we consider the reverse process (see

Fig. 6.3 (b)). We insert the partition in the middle in an (N + M)-particle gas. The

number of particles in the left side, n, after the reinsertion must be N to recover the

original state. This is the only non-singular event. Therefore, the singular probability is

λS = 1− N+MCN2N+M

. (6.50)

Thus, from Eq. (6.44), the thermodynamic entropy production is

∆S = (N +M)kB ln 2− kB lnN+MCN . (6.51)

On the other hand, from Eq. (6.31) and the additivity of the thermodynamic entropy, we

obtain

∆S = (N +M)kB ln 2 + kBfC(N +M)− kBf

C(N)− kBfC(M) (6.52)

117


fC(N +M)− fC(N)− fC(M) = − lnN+MCN . (6.53)

When set M = 1, we have

fC(N + 1)− fC(N)− fC(1) = − ln(N + 1). (6.54)

Let us define

gC(N) = exp[fC(N)], (6.55)

and then Eq. (6.54) reduces to

gC(N + 1) = gC(N)gC(1)

N + 1. (6.56)

This equation can be rewritten as

(N + 1)! · gC(N + 1) = gC(1) ·N ! · gC(N). (6.57)


N ! · gC(N) = gC(1)N (6.58)

and

gC(N) =gC(1)N

N !. (6.59)

Hence, we conclude

fC(N) = NfC(1)− lnN !. (6.60)

Thus, the desired factor N ! is reproduced.

In summary, based on our nonequilibrium equality that is applicable in the presence

of absolute irreversibility, we have shown that the difference of the entropy productions

in the two mixing processes originates from the difference of the degree of absolute irre-

versibility. Furthermore, we have reproduced the factor N ! in the relation between the

thermodynamic entropy and the classical statistical mechanical entropy. Our new reso-

lution automatically takes account of the sub-leading term and mesoscopic effects in the

118

thermodynamic entropy, which was ignored in the resolution based on the extensivity of

the thermodynamic entropy.

119

Chapter 7

Conclusions and Future Prospects

7.1 Conclusions

In this thesis, we have investigated the situations to which the conventional integral

nonequilibrium equalities cannot apply, and proposed a new concept of absolute irre-

versibility to describe these situations in a unified manner. In absolutely irreversible pro-

cesses, some of time-reversed paths have no counterpart in the original forward process,

and the entropy production diverges in the context of the detailed fluctuation theorems.

In mathematical terms, absolute irreversibility is defined as the singular part of the time-

reversed probability measure with respect to the forward probability measure. Lebesgue’s

decomposition enables us to separate the absolutely irreversible part from the ordinar-

ily irreversible part. As a consequence, we have obtained the integral nonequilibrium

equalities in the presence of absolute irreversibility. The obtained equalities involve two

physical quantities related to irreversibility: the entropy production representing ordinary

irreversibility and the singular probability describing absolute irreversibility. The corre-

sponding inequalities give tighter fundamental restrictions on the entropy production in

nonequilibrium processes than the conventional second-law like inequalities. Our nonequi-

librium equalities have been verified in free expansion and in numerical simulations of the

two Langevin systems.

Moreover, we have generalized our nonequilibrium equalities in absolutely irreversible

processes to the situations in which the system is subject to measurement-based feedback

control. We have transformed the obtained nonequilibrium equalities and introduced

a concept of unavailable information, which characterizes the inevitable inefficiency of

feedback protocols. As a result, we have derived inequalities that give an achievable

lower bound of the entropy production. We have verified our information-thermodynamic

absolutely irreversible nonequilibrium equalities in the process with a measurement and

trivial feedback control and in the two- and multi-particle Szilard engines.

120

We have applied the notion of absolute irreversibility to the gas-mixing problem of

Gibbs’ paradox. The difference between the entropy production of the mixing process of

two different gases and that of two identical gases originates from the difference of absolute

irreversibility, i.e., different behaviors under the reinsertion of the partition. Moreover, we

have reproduced the factorial in the particle-number dependence of the thermodynamic

entropy. Our quantitative resolution of Gibbs’ paradox applies to a classical mesoscopic

regime, where Pauli’s resolution based on the extensivity of the thermodynamic entropy

breaks down.

7.2 Future prospects

As future prospects, I enumerate several outstanding issues.

First of all, I intend to generalize our absolutely irreversible integral nonequilibrium

equalities to the quantum regime. A part of this extension has already been done, and

we have shown that absolute irreversibility is essential under inefficient feedback control

and projective measurements [74]. However, nonequilibrium equalities under the condition

that the initial state has such quantum correlations as entanglement are elusive. Absolute

irreversibility may play an important role in these quantum nonequilibrium situations.

Next, I intend to apply our formulation based on measure theory to the chaotic sys-

tems. Although the nonequilibrium equalities were first proven in chaotic systems, most

of recent researches of nonequilibrium equalities are restricted to such simple systems as

the Langevin systems. I expect our formulation is compatible with the chaotic system

that has a singular continuous probability measure with respect to the Lebesgue mea-

sure. This issue may be related to a topic of thermalization because the relaxed state

after a nonequilibrium process starting from the canonical distribution sometimes exhibits

singular behaviors.

Finally, I contemplate applying our nonequilibrium equality to finite-time thermody-

namics. Finite-time thermodynamics is a field of thermodynamics that puts emphasis on

power of thermodynamic engines and seeks to their optimal efficiency. Therefore, thermo-

dynamic engines under consideration are subject to finite-time nonequilibrium processes.

Recently, in the context of nonequilibrium equalities, the efficiency of finite-time engines

has been studied [75]. I expect that our nonequilibrium equalities with absolute irre-

versibility give new restrictions on the efficiency.

121

Appendix A

From the Langevin Dynamics to

Other Formulations

In this Appendix, we derive the path integral formula (2.39) and the Fokker-Planck equa-

tion (2.41) from the overdamped Langevin equation

x(t) = µF (x(t), λ(t)) + ζ(t), (A.1)

where ζ(t) is a white Gaussian noise satisfying

〈ζ(t)〉 = 0, (A.2)

〈ζ(t)ζ(t′)〉 = 2Dδ(t− t′). (A.3)

A.1 Path-integral formula

To calculate the path probability, we first discretize the time interval [0, τ ] into N sections

with the same length ∆t = τ/N . We define ti = i∆t (i = 0, 1, · · · , N), xi = x(ti), and

λi = λ(ti). Then, the discretized Langevin equation reads1

xi+1 − xi = µF (xi, λi) + F (xi+1, λi+1)

2∆t+ ∆Wi+1, (A.4)

where

∆Wi+1 =

∫ ti+1

ti

ζ(t)dt (A.5)

1We use the Stratonovich convention.

122

for i = 0, · · · , N −1. The discretized noise ∆Wi obeys the Gaussian distribution with the

average

〈∆Wi〉 =

∫ ti

ti−1

〈ζ(t)〉dt = 0 (A.6)

and the variance

〈∆W 2i 〉 =

∫ ti

ti−1

∫ ti

ti−1

〈ζ(t)ζ(t′)〉dtdt′

=

∫ ti

ti−1

∫ ti

ti−1

2Dδ(t− t′)dtdt′

=

∫ ti

ti−1

2Ddt

= 2D∆t. (A.7)

Therefore, the probability distribution of ∆Wi is given by

P (∆Wi) =1√

4πD∆texp

[−∆W 2

i

4D∆t

]. (A.8)

Thus, the joint probability distribution of all ∆Wi is calculated as

P [∆W] =1

(√

4πD∆t)nexp

[− 1

4D∆t

N∑i=1

∆W 2i

]

=1

(√

4πD∆t)nexp

[− 1

4D

N−1∑i=0

(xi+1 − xi

∆t− µF (xi, λi) + F (xi+1, λi+1)

2

)2

∆t

].

(A.9)

In the large-N limit, we obtain

P [∆W] = N exp

[− 1

4D

∫ τ

0

(x(t)− µF (x(t), λ(t)))2 dt

], (A.10)

where N is the normalization constant.

The path probability P [x|x0] is defined in terms of the probability distribution

P [∆W] as

P [x|x0]N∏i=1

dxi = P [∆W]N∏i=1

d∆Wi. (A.11)

123

To obtain the path probability, we calculate the Jacobian as

J = det

(∂∆Wi

∂xj

)= det

(δi,j

(1− µ

2∂xF (xi, λi)∆t

)− δi−1,j

(1 +

µ

2∂xF (xi−1, λi−1)∆t

))=

N∏i=1

(1− µ


)=

N∏i=1

exp[−µ


]= exp

[−

N∑i=1

µ


]. (A.12)

Thus, in the continuous limit (∆t→ 0), we obtain

J = exp

[−∫ τ

0

µ

2∂xF (x(t), λ(t))dt

](A.13)

and therefore

P [x|x0] = P [∆W]J

= N exp

[−∫ τ

0

((x(t)− µF (x(t), λ(t)))2

4D+µ

2∂xF (x(t), λ(t))

)dt

],

(A.14)

which is nothing but the path integral formula for the overdamped Langevin equation

(2.39).

A.2 Fokker-Planck equation

First of all, we evaluate the time evolution of an arbitrary function f(x(t)) as

df(x(t)) = f(x(t+ dt))− f(x(t))

= f ′(x(t))dx+1

2f ′′(x(t))dx2 +O((dx)3), (A.15)

where

dx = x(t+ dt)− x(t) = µF (x(t), λ(t))dt+

∫ t+dt

t

ζ(t′)dt′ +O((dt)2). (A.16)

124

We take the statistical average of Eq. (A.15). Substituting Eq. (A.16) into the average of

the first term on the right-hand side of Eq. (A.15), we obtain

〈f ′(x(t))dx〉 = 〈µF (x(t), λ(t))f ′(x(t))〉dt+

∫ t+dt

t

〈f ′(x(t))ζ(t′)〉dt′ +O((dt)2). (A.17)

Since the stochastic quantity f ′(x(t)) at time t is independent of the noise ζ(t′) for t′ > t,

we obtain

〈f ′(x(t))ζ(t′)〉 = 〈f ′(x(t))〉〈ζ(t′)〉 = 0 (t′ > t). (A.18)

Therefore, Eq. (A.17) reduces to

〈f ′(x(t))dx〉 = 〈µF (x(t), λ(t))f ′(x(t))〉dt+O((dt)2). (A.19)

In a similar way, the average of the second term on the right-hand side of Eq. (A.15)

reduces to

1

2〈f ′′(x(t))dx2〉 =

1

2〈µ2F (x(t), λ(t))2f ′′(x(t))〉dt2 +

∫ t+dt

t

〈µF (x(t), λ(t))f ′′(x(t))ζ(t′)〉dt

+1

2

∫ t+dt

t

∫ t+dt

t

〈ζ(t′)ζ(t′′)f ′′(x(t))〉dt′dt′′ +O((dt)3)

= D

∫ t+dt

t

∫ t+dt

t

δ(t′ − t′′)〈f ′′(x(t))〉dt′dt′′ +O((dt)2)

= D〈f ′′(x(t))〉dt+O((dt)2). (A.20)

Therefore, the average of Eq. (A.15) is

d〈f(x(t))〉 = 〈µF (x(t), λ(t))f ′(x(t))〉dt+D〈f ′′(x(t))〉dt+O((dt)2). (A.21)

To derive the Fokker-Planck equation, we note that the probability p(x, t) to find the

Langevin particle at position x at time t is given by

p(x, t) = 〈δ(x− x(t))〉. (A.22)

If we set f(x(t)) = δ(x− x(t)), Eq. (A.21) reduces to

dp(x, t) = −〈µF (x(t), λ(t))δ′(x− x(t))〉dt+D〈δ′′(x− x(t))〉dt+O((dt)2)

= −∂x〈µF (x(t), λ(t))δ(x− x(t))〉dt+D∂2x〈δ(x− x(t))〉dt+O((dt)2)

= −∂x[µF (x, λ(t))p(x, t)]dt+D∂2xp(x, t)dt+O((dt)2). (A.23)

125

Thus, we obtain the following Fokker-Planck equation

∂tp(x, t) = −∂x[(µF (x, λ(t))−D∂x)p(x, t)]. (A.24)

126

Appendix B

Measure Theory and Lebesgue’s

Decomposition

In this Appendix, we briefly review measure theory and Lebesgue’s decomposition theo-

rem. This Appendix is mainly based on [26].

B.1 Preliminary subjects

Definition 1 (σ-algebra) A family X of subsets of X is said to be a σ-algebra if the

following three conditions are met:

(i) ∅ and X belong to X ;

(ii) If A belongs to X, then X\A belongs to X ;

(iii) If (An) is a sequence of sets in X , then⋃∞n=1 An belongs to X .

Definition 2 (measurable space) An ordered pair (X,X ) consisting of a set X and a

σ-algebra X of subsets of X is called a measurable space.

Definition 3 (measure) A measure is an extended real-valued function µ defined on

a σ-algebra X of subsets of X satisfying the following conditions:

(i) µ(∅) = 0;

(ii) µ(E) ≥ 0 for all E ∈ X ;

(iii) µ is countably additive, i.e., if (En) is any disjoint sequence of sets in X , then

µ

(∞⋃n=1

En

)=∞∑n=1

µ(En). (B.1)

127

Definition 4 (measure space) A measure space is an ordered triad (X,X , µ) con-

sisting of a set X, a σ-algebra X of subsets of X and a measure µ defined on X .

Definition 5 ((σ-)finite measure) Let (X,X , µ) be a measure space. If µ does not

take on an infinite value, we say that µ is finite. If there exists a sequence (En) of sets

in X with X =⋃∞n=1En and µ(En) <∞ for all n, then we say µ is σ-finite.

Definition 6 (mesurable function) An R-valued function f with domain X is said to

be X -measurable if for any real number α the set

x ∈ X |f(x) > α (B.2)

belongs to X .

Definition 7 (almost everywhere) Let µ be a measure on X . A certain proposition is

said to hold µ-almost everywhere on X if there exists a subset N ∈ X with µ(N) = 0

such that the proposition holds on X\N .

Theorem 1 Suppose that f is a nonnegative X -measurable function. Then, f(x) = 0

µ-almost everywhere on X iff ∫fdµ = 0. (B.3)

B.2 Classification of measures

Definition 8 (absolutely continuous) A measure ν on X is said to be absolutely

continuous with respect to a measure µ on X if E ∈ X and µ(E) = 0 imply ν(E) = 0.

In this case, we write ν µ.

Lemma 1 Let µ and ν be finite measures on X . Then ν µ iff for every ε > 0 there

exists a δ > 0 such that E ∈ X and µ(E) < δ imply that ν(E) < ε.

Proof. If this condition is satisfied and µ(E) = 0, then ν(E) < ε for all ε > 0, which

implies ν(E) = 0.

Conversely, suppose that there exist some ε > 0 and En ∈ X with µ(En) < 2−n and

ν(En) ≥ ε. Let Fn =⋃∞k=nEk, so that µ(Fn) < 2−n+1 and ν(Fn) ≥ ε. Since (Fn) is a

decreasing sequence of measurable sets and µ, ν are finite measures, we have

µ

(∞⋂n=1

Fn

)= lim

n→∞µ(Fn) = 0, ν

(∞⋂n=1

Fn

)= lim

n→∞ν(Fn) ≥ ε. (B.4)

128

Therefore, ν is not absolutely continuous with respect to µ.

Intuitively, ν µ means that a set that has a small µ-measure also has a small

ν-measure.

Definition 9 (singular) Two measures µ and ν on X are said to be mutually singular

if there are sets A and B ∈ X that satisfy X = A ∪B, ∅ = A ∩B and µ(A) = ν(B) = 0.

In this case, we write µ ⊥ ν.

Despite this symmetric definition, we also say that ν is singular with respect to µ.

Lemma 2 Let α be a measure such that α µ and α ⊥ µ, then α = 0.

Proof. Since α ⊥ µ, there exist sets A and B such that

X = A ∪B, ∅ = A ∩B, α(A) = 0, µ(B) = 0. (B.5)

Since α µ and µ(B) = 0, α(B) = 0. Then, due to the additivity of α, we have

α(X) = α(A) + α(B) = 0. (B.6)

It follows that for all E ∈ X

0 ≤ α(E) = α(X)− α(X\E) ≤ 0, (B.7)

which implies α = 0.

Definition 10 (discontinuous point) For ω ∈ X, if µ(ω) > 0, then ω is said to be

a discontinuous point of µ.

Definition 11 (discrete measure) Let C denote the set of all the discontinuous points

of µ. Then, µ is said to be discrete measure if µ(X) = µ(C).

Definition 12 (continuous measure) A measure µ is said to be a continuous mea-

sure if µ has no discontinuous points.

B.3 Radon-Nikodym theorem

Theorem 2 (Radon-Nikodym theorem) Let µ and ν be σ-finite measures defined on

X and suppose that ν is absolutely continuous with respect to µ. Then, there exists a

129

nonnegative X -measurable function f such that

ν(E) =

∫E

fdµ, ∀E ∈ X . (B.8)

Moreover, the function f is uniquely determined µ-almost everywhere.

The function f is referred to as the Radon-Nikodym derivative, and formally written

as

f =dν

dµ. (B.9)

Moreover, f is the transformation function from ν to µ. In fact, for an arbitrary X -

measurable function g, we have∫E

gdν =

∫E

gdν

dµdµ =

∫E

gfdµ, ∀E ∈ X . (B.10)

B.4 Lebesgue’s decomposition theorem

Theorem 3 (Lebesgue’s decomposition theorem 1) Let µ and ν be σ-finite mea-

sures defined on a σ-algebra X . Then there exist measures νAC and νS such that ν =

νAC + νS, νAC µ and νS ⊥ µ. Moreover, the measures νAC and νS are unique.

Proof. Let λ = µ + ν. Then, µ and ν are absolutely continuous with respect to λ.

Therefore, we can apply the Radon-Nikodym theorem to obtain

µ(E) =

∫E

fdλ, ν(E) =

∫E

gdλ (B.11)

for all E ∈ X , where f , g are nonnegative X -measurable functions. Let A = x|f(x) > 0and B = x|f(x) = 0 so that A ∩B = ∅ and X = A ∪B.

Define νAC and νS for E ∈ X by

νAC(E) = ν(E ∩ A), νS(E) = ν(E ∩B). (B.12)

Since ν is additive, ν(E) = νAC(E) + νS(E). To see νAC µ, we note that if µ(E) = 0,

then ∫E

fdλ = 0. (B.13)

130

In accordance with Theorem 1, f(x) = 0 for λ-almost all x ∈ E, which means λ(E∩A) = 0.

Since ν λ, ν(E ∩ A) = 0, and then νAC(E) = 0. Thus, νAC is absolutely continuous

with respect to µ. On the other hand, since νS(A) = µ(B) = 0, νS is singular with respect

to µ.

The uniqueness of this decomposition can be established by Lemma 2.

Lemma 3 Let ν be a measure defined on a σ-algebra X . Then there exist measures νc

and νd such that ν = νc + νd, where νc is continuous and νd is discrete. Moreover, the

measures νc and νd are unique.

Proof. Let C be the set of all the discontinuous points of ν. Define νc and νd for E ∈ Xby

νc(E) = ν(E\C), νd(E) = ν(E ∩ C). (B.14)

Then, we can show that νc is continuous and νd is discrete.

Suppose

νc + νd = ν ′c + ν ′d, (B.15)

where ν ′c is continuous and ν ′d is discrete. If νd 6= ν ′d, there exists a single point ω ∈ X such

that νd(ω) 6= ν ′d(ω). On the other hand, due to the continuity, νc(ω) = ν ′c(ω) = 0.

These relations lead to

νc(ω) + νd(ω) 6= ν ′c(ω) + ν ′d(ω), (B.16)

which contradicts Eq. (B.15). Hence, νd = ν ′d and therefore the uniqueness of the decom-

position is established.

To introduce a stronger version of Lebesgue’s decomposition, we prove the following

lemma.

Lemma 4 Let µ and ν be measures on a σ-algebra. If µ is continuous and ν is discrete,

then ν is singular with respect to µ.

Proof. Let C denote the set of all the discontinuous points of ν. Then, ν(X\C) = 0. On

the other hand, since µ is continuous and C is countable, µ(C) = 0. Thus, ν and µ are

mutually singular.

131

Theorem 4 (Lebesgue’s decomposition theorem 2) Let µ and ν be σ-finite mea-

sures defined on a σ-algebra X and suppose µ is continuous. Then there exist measures

νac, νsc and νd such that ν = νac + νsc + νd, where νac µ; νsc ⊥ µ and νsc is continuous;

νd is discrete. Moreover, the measures νac, νsc and νd are unique.

Proof. We can apply Theorem 3 to uniquely decompose

ν = νac + νs, (B.17)

where νac is absolutely continuous with respect to µ, and νs is singular with respect to µ.

In accordance with Lemma 3, we can uniquely decompose νs into two parts:

νs = νsc + νd, (B.18)

where νsc is continuous and νd is discrete. Thus, we obtain the following decomposition:

ν = νac + νsc + νd. (B.19)

The uniqueness of this decomposition follows from Lemma 4.

The continuity of µ is needed to establish the uniqueness of the decomposition. If µ

has a discontinuous point ω ∈ X, then the measure νω that satisfies νω(ω) = νω(X) is

absolutely continuous with respect to µ and discrete at the same time.

132

Acknowledgement

The studies in this thesis was done when the author was a master-course student in

Masahito Ueda group in the University of Tokyo. This thesis would not be completed

without help of a lot of collaborators and colleagues.

First of all, I would like to express my best gratitude to my supervisor, Prof. Masahito

Ueda. He made me cognizant of the topic in this thesis and gave me a lot of insightful

comments throughout discussions. He also read the manuscript with great attention and

gave me enormous suggestions. I would also thank him for the best research environment

that he arranged for me.

I would also like to thank my collaborator, Ken Funo, for fruitful discussions on our

studies and for teaching me quantum aspects of nonequilibrium equalities.

I am thankful for my collaborator, Yuto Ashida, for his constructive ideas from his

deep comprehension of our field.

I am grateful to Prof. Takahiro Sagawa for his critical comments and beneficial dis-

cussions on our work.

I appreciate critical comments and constructive suggestions by Prof. Shin-ichi Sasa.

I acknowledge comments from anonymous referees of our article [27], which I find very

useful to clarify physical meanings of our work.

I would like to express my deep sense of gratitude to the members in Masahito Ueda

group, especially to Yui Kuramochi and Tomohiro Shitara for fruitful discussions on math-

ematical aspects of our work, and to Tatsuhiko N. Ikeda for suggestive comments.

I am also indebted to the members in Masaki Sano group in the University of Tokyo,

especially to Kyogo Kawaguchi for lecturing me on theoretical aspects of mesoscopic

physics and to Yohei Nakayama, Yuta Hirayama and Daiki Nishiguchi for teaching me

experimental techniques of this field.

Finally, I acknowledge financial support from the Japan Society for the Promotion of

Science (JSPS) through the Program for Leading Graduate Schools (MERIT).

133

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